1 Abstract

The muHVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data analysis. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below:

Data Compression: Vector quantization (VQ), HVQ (hierarchical vector quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective.
Data Projection: Dimension projection of the compressed cells to 1D,2D or 3D with the Sammons Non-linear Algorithm. This step creates topology preserving map (also called as embedding) coordinates into the desired output dimension.
Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for hierarchical Voronoi tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks.
Prediction: Scoring new data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required.

X	price	speed	hd	ram	screen	cd	multi	premium	ads	trend
1	1499	25	80	4	14	no	no	yes	94	1
2	1795	33	85	2	14	no	no	yes	94	1
3	1595	25	170	4	15	no	no	yes	94	1
4	1849	25	170	8	14	no	no	no	94	1
5	3295	33	340	16	14	no	no	yes	94	1
6	3695	66	340	16	14	no	no	yes	94	1

Row.No	price	speed	hd	ram	screen	ads
2463	2799	50	230	8	15	216
2511	2197	33	270	4	14	216
2227	2744	50	340	8	17	275
526	2999	66	245	16	15	139
4291	1974	33	200	4	14	248
2986	2490	33	528	16	14	267

3 Map A : Base Compressed Map

Let us try to visualize the compressed Map A from the flow diagram below.

Figure 1: Flow map with highlighted bounding box in red around compressed map A

This package can perform vector quantization using the following algorithms -

Hierarchical Vector Quantization using k−means
Hierarchical Vector Quantization using k−medoids

For more information on vector quantization, refer the following link.

The HVT function constructs highly compressed hierarchical Voronoi tessellations. The raw data is first scaled and this scaled data is supplied as input to the vector quantization algorithm. The vector quantization algorithm compresses the dataset until a user-defined compression percentage/rate is achieved using a parameter called quantization error which acts as a threshold and determines the compression percentage. It means that for a given user-defined compression percentage we get the ‘n’ number of cells, then all of these cells formed will have a quantization error less than the threshold quantization error.

Let’s try to comprehend the HVT function first before moving ahead.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  normalize = TRUE,
  diagnose = FALSE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters have been explained below :

dataset - A dataframe with numeric columns.
min_compression_perc - An integer indicating the minimum percent compression rate to be achieved for the dataset.
n_cells - An integer indicating the number of cells per hierarchy (level).
depth - An integer indicating the number of levels. (1 = No hierarchy, 2 = 2 levels, etc …).
quant.error - A number indicating the quantization error threshold. A cell will only breakdown. into further cells if the quantization error of the cell is above the defined quantization error threshold.
projection.scale - A number indicating the scale factor for the tesselations so as to visualize the sub-tesselations well enough.
scale_summary - A list with mean and standard deviation values for all the features in the dataset. Pass the scale summary when the input dataset is already scaled or normalize is set to False.
distance_metric - The distance metric can be L1_Norm or L2_Norm. L1_Norm is selected by default. The distance metric is used to calculate the distance between an n dimensional point and centroid. The user can also pass a custom function to calculate this distance.
error_metric - The error metric can be mean or max. max is selected by default. max will return the max of m values and mean will take mean of m values where each value is a distance between a point and centroid of the cell. Moreover, the user can also pass a custom function to calculate the error metric.
quant_method - The quantization method can be kmeans or kmedoids. kmeans is selected by default.
normalize - A logical value indicating whether the columns in your dataset need to be normalized. Default value is TRUE. The algorithm supports Z-score normalization.
diagnose - A logical value indicating whether user wants to perform diagnostics on the model. Default value is TRUE.
hvt_validation - A logical value indicating whether user wants to holdout a validation set and find mean absolute deviation of the validation points from the centroid. Default value is FALSE.
train_validation_split_ratio - A numeric value indicating train validation split ratio. This argument is only used when hvt_validation has been set to TRUE. Default value for the argument is 0.8.

We will use the HVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the HVT function again.

In our example we will iteratively increase the number of cells until the desired compression percentage is reached instead of increasing the quantization threshold because it may reduce the level of detail captured in the data representation

First, we will construct map A by using the below mentioned model parameters.

3.0.1 Iteration 1:

We will pass the below mentioned model parameters along with training dataset to HVT function.

Model Parameters

Number of Cells at each Level = 200
Maximum Depth = 1
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

set.seed(240)
map_A <- list()
map_A  <- muHVT::HVT(
  trainComputers,
  n_cells = 200,
  depth = 1,
  quant.err = 0.2,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary.

compressionSummaryTable(map_A[[3]]$compression_summary)

segmentLevel	noOfCells	noOfCellsBelowQuantizationError	percentOfCellsBelowQuantizationErrorThreshold	parameters
1	200	83	0.42	n_cells: 200 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 42% of cells have reached the quantization threshold error. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if desired compression (80%) is reached

3.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

Number of Cells at each Level = 440
Maximum Depth = 1
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

map_A <- list()
map_A <-muHVT::HVT(trainComputers,
                n_cells = 440,
                quant.err = 0.2,
                depth = 1,
                distance_metric = "L1_Norm",
                error_metric = "max",
                quant_method = "kmeans",
                normalize = T)

As per the manual, map_A[[3]] gives us detailed information about the hierarchical vector quantized data. map_A[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from map A

summaryTable(map_A[[3]]$summary,scroll = T,limit = 500)

Segment.Level	Segment.Parent	Segment.Child	n	Cell.ID	Quant.Error	price	speed	hd	ram	screen	ads
1	1	1	7	46	0.08	-0.76	-0.89	-0.88	-0.76	-0.67	1.57
1	1	2	10	108	0.08	-0.80	-0.89	-0.16	-0.76	-0.67	0.67
1	1	3	15	223	0.12	0.37	-0.89	-0.72	-0.05	0.43	-1.65
1	1	4	11	54	0.07	-1.50	-0.89	-0.75	-0.76	-0.67	0.62
1	1	5	8	146	0.13	-0.31	0.68	-0.95	-0.89	-0.67	-0.14
1	1	6	11	150	0.16	-0.66	0.68	-0.78	-0.79	-0.67	-0.73
1	1	7	11	170	0.1	0.03	-1.24	-0.13	-0.05	-0.67	0.38
1	1	8	8	334	0.15	0.62	2.30	0.08	-0.05	0.43	0.04
1	1	9	8	114	0.07	-0.16	0.68	-1.19	-1.11	-0.67	0.87
1	1	10	7	248	0.17	0.51	-0.08	0.34	-0.05	-0.67	-0.30
1	1	11	9	140	0.12	-0.01	0.68	-1.15	-1.00	-0.67	0.33
1	1	12	7	219	0.14	-1.36	0.24	0.46	-0.05	-0.67	-0.74
1	1	13	9	271	0.05	-1.08	0.68	0.49	-0.05	0.43	-0.84
1	1	14	19	109	0.06	-0.31	-0.89	-0.74	-0.76	-0.67	0.38
1	1	15	6	176	0.08	-0.72	-0.89	-0.07	-0.05	0.43	0.72
1	1	16	17	332	0.14	0.42	2.30	0.10	-0.05	0.43	1.50
1	1	17	12	18	0.05	-1.21	-1.27	-1.19	-1.11	-0.67	0.97
1	1	18	19	149	0.16	-0.68	-0.08	-0.46	-0.76	0.43	0.79
1	1	19	17	428	0.35	0.18	2.30	2.53	1.37	0.43	-2.22
1	1	20	20	320	0.36	0.82	-0.16	-0.09	-0.12	2.64	0.71
1	1	21	3	305	0.18	2.27	-0.35	-0.01	-0.05	-0.67	-1.32
1	1	22	7	227	0.1	-0.51	-0.89	0.45	-0.05	0.43	-0.47
1	1	23	10	178	0.12	0.00	0.68	-0.86	-0.76	-0.67	-0.90
1	1	24	9	365	0.1	0.68	-0.08	1.20	1.37	0.43	-0.36
1	1	25	5	14	0.11	-1.99	-0.89	-0.96	-1.11	-0.67	0.18
1	1	26	3	411	0.05	1.25	-0.89	2.29	2.79	0.43	0.57
1	1	27	18	122	0.15	-0.18	-0.98	-0.85	-0.76	0.43	0.68
1	1	28	15	189	0.11	0.40	-0.92	0.03	-0.05	-0.67	0.87
1	1	29	11	107	0.11	-0.49	-0.96	-0.88	-0.76	-0.67	-0.64
1	1	30	7	423	0.47	3.55	0.12	2.51	1.37	-0.67	0.44
1	1	31	14	90	0.05	-0.63	-0.89	-0.79	-0.76	-0.67	0.58
1	1	32	22	430	0.24	0.63	0.75	3.07	2.79	0.43	-2.27
1	1	33	5	390	0.3	1.37	-0.89	3.73	-0.19	-0.45	0.70
1	1	34	25	101	0.18	-0.85	-0.97	-0.71	-0.76	0.43	0.84
1	1	35	11	425	0.07	0.15	2.30	1.70	1.37	0.43	-2.39
1	1	36	10	358	0.05	0.24	-0.89	1.20	1.37	0.43	-0.84
1	1	37	16	166	0.11	0.03	0.68	-1.08	-0.78	-0.67	-1.65
1	1	38	13	45	0.05	-0.91	-0.89	-1.19	-1.11	-0.67	0.42
1	1	39	8	383	0.12	1.15	2.30	0.45	1.37	-0.67	-0.16
1	1	40	5	9	0.07	-1.24	-0.97	-1.19	-1.11	-0.67	1.57
1	1	41	11	419	0.06	1.41	-0.89	2.29	2.79	0.43	-0.82
1	1	42	8	242	0.14	-0.81	-0.08	0.30	-0.05	0.43	-0.65
1	1	43	13	179	0.09	0.41	0.68	-0.76	-0.76	-0.67	0.40
1	1	44	5	375	0.04	0.06	-0.08	1.70	1.37	0.43	-0.79
1	1	45	20	129	0.14	-1.15	0.68	-0.79	-0.76	-0.67	-0.40
1	1	46	10	292	0.22	0.86	0.68	-0.65	-0.12	0.43	-1.41
1	1	47	5	79	0.12	-0.89	-1.04	-0.94	-0.05	-0.67	1.02
1	1	48	23	246	0.11	-0.42	0.68	0.46	-0.05	-0.67	-0.63
1	1	49	8	207	0.25	0.74	-0.89	-0.40	-0.40	0.43	0.52
1	1	50	11	27	0.06	-1.06	-1.27	-1.19	-1.11	-0.67	0.43
1	1	51	8	51	0.11	-1.23	-0.08	-1.05	-0.85	-0.67	0.94
1	1	52	19	288	0.09	0.81	-0.89	0.45	1.37	-0.67	0.88
1	1	53	7	154	0.1	-0.62	0.68	-0.15	-0.76	-0.67	0.84
1	1	54	10	261	0.15	0.61	-0.08	-0.67	-0.05	0.43	-1.40
1	1	55	10	195	0.15	0.18	0.77	-0.09	-0.76	-0.67	0.83
1	1	56	14	250	0.09	0.52	0.68	-0.69	-0.05	-0.67	-1.61
1	1	57	20	331	0.2	-0.63	0.68	0.30	-0.76	2.64	-0.95
1	1	58	9	379	0.15	1.33	-0.08	-0.65	-0.29	2.64	-1.52
1	1	59	14	11	0.21	-0.28	-1.05	-0.79	-0.76	2.64	0.53
1	1	60	29	359	0.13	1.36	0.68	0.18	1.37	0.43	0.77
1	1	61	6	337	0.1	2.46	0.68	0.21	-0.05	-0.67	-0.87
1	1	62	6	1	0.17	-0.17	-1.21	-1.02	-0.76	2.64	1.32
1	1	63	28	243	0.33	-0.33	2.30	-0.23	-0.46	-0.67	-0.88
1	1	64	8	274	0.07	0.41	-1.27	0.45	1.37	-0.67	0.68
1	1	65	13	362	0.14	1.07	0.75	0.35	1.37	0.43	1.34
1	1	66	10	143	0.07	-0.34	-0.89	-0.80	-0.05	-0.67	-1.66
1	1	67	4	265	0.05	-0.55	0.68	1.23	-0.05	-0.67	-0.69
1	1	68	11	13	0.15	-0.83	-0.89	-0.25	-0.76	2.64	-0.33
1	1	69	8	298	0.17	-0.62	0.20	2.29	-0.05	-0.67	-0.95
1	1	70	4	335	0.06	1.34	-0.08	0.45	1.37	-0.67	-0.08
1	1	71	20	204	0.16	0.09	-0.08	0.02	-0.05	-0.67	0.86
1	1	72	10	42	0.06	-1.49	-0.89	-0.75	-0.76	-0.67	1.04
1	1	73	1	429	0	3.08	0.68	0.04	4.20	0.43	0.71
1	1	74	14	186	0.14	-0.79	-0.89	0.45	-0.05	-0.67	-0.68
1	1	75	4	410	0.37	2.27	0.68	3.73	-0.23	-0.40	0.68
1	1	76	9	163	0.16	1.05	-0.89	-0.41	-0.60	-0.67	0.61
1	1	77	10	400	0.07	-0.03	0.68	1.70	1.37	0.43	-2.38
1	1	78	6	275	0.18	1.14	0.68	0.13	-0.05	-0.67	-0.18
1	1	79	25	241	0.16	-0.88	0.68	0.40	-0.05	-0.67	-1.06
1	1	80	6	245	0.14	-1.22	0.68	-0.30	-0.05	0.43	-0.91
1	1	81	21	120	0.16	-0.46	-0.08	-0.82	-0.62	-0.67	0.70
1	1	82	11	40	0.18	-0.93	-0.99	-1.19	-1.11	0.43	0.37
1	1	83	9	342	0.28	1.16	0.43	-0.52	-0.68	2.64	1.12
1	1	84	8	286	0.05	-0.72	1.11	0.50	-0.05	0.43	-0.83
1	1	85	8	33	0.1	-1.06	-0.89	-1.18	-1.02	-0.67	-0.99
1	1	86	5	282	0.23	-1.50	0.53	0.19	-0.48	0.43	-2.16
1	1	87	17	137	0.07	-0.31	0.68	-0.78	-0.76	-0.67	0.96
1	1	88	19	168	0.07	-0.08	-0.89	0.05	-0.05	-0.67	1.04
1	1	89	7	291	0.05	1.10	-0.89	0.15	1.37	-0.67	0.36
1	1	90	6	24	0.19	-0.97	-1.02	-1.07	-0.94	0.43	-1.40
1	1	91	4	434	0.59	4.07	1.49	1.29	0.30	2.64	0.07
1	1	92	19	409	0.65	0.92	0.42	1.41	1.37	2.64	-0.60
1	1	93	9	393	0.46	2.04	2.30	0.91	-0.05	-0.31	-0.34
1	1	94	8	22	0.08	-1.66	-1.27	-0.84	-0.76	-0.67	0.84
1	1	95	23	158	0.22	-1.41	0.68	-0.20	-0.70	-0.67	-1.05
1	1	96	11	330	0.22	1.55	0.68	-0.45	-0.31	0.43	-1.59
1	1	97	6	145	0.17	0.49	-0.89	-0.64	-0.76	-0.67	-0.17
1	1	98	12	121	0.15	0.21	-0.89	-0.80	-0.76	-0.67	-1.70
1	1	99	14	329	0.24	2.06	0.63	0.31	-0.25	0.43	0.99
1	1	100	16	299	0.05	1.21	-0.89	0.46	1.37	-0.67	0.86
1	1	101	5	328	0.1	-0.87	1.11	1.33	-0.05	0.43	-1.13
1	1	102	10	278	0.1	0.33	-0.93	0.45	1.37	-0.67	0.02
1	1	103	5	102	0.1	-1.05	0.68	-0.76	-0.76	-0.67	1.27
1	1	104	12	25	0.08	-1.25	-1.05	-0.78	-0.76	-0.67	1.57
1	1	105	9	385	0.25	2.05	0.34	-0.21	-0.05	2.64	1.12
1	1	106	10	231	0.18	-0.02	-0.08	0.02	-0.05	0.43	1.29
1	1	107	5	193	0.14	-0.66	-0.08	-0.33	-0.05	-0.67	-1.01
1	1	108	4	418	0.06	-0.03	0.68	3.07	1.37	0.43	-2.25
1	1	109	4	306	0.16	1.83	0.68	-0.59	-0.05	-0.67	-1.57
1	1	110	5	378	0.15	1.09	0.68	1.20	1.37	0.43	-0.11
1	1	111	2	308	0.07	-0.55	0.68	0.26	1.37	-0.67	-1.25
1	1	112	18	239	0.15	0.44	0.75	0.08	-0.05	-0.67	1.05
1	1	113	11	62	0.06	-1.14	-0.89	-0.81	-0.76	-0.67	0.73
1	1	114	5	169	0.07	-0.26	-0.89	0.47	-0.05	-0.67	1.40
1	1	115	18	415	0.11	0.74	-0.08	2.29	2.79	0.43	-0.93
1	1	116	5	97	0.13	0.23	-1.19	-1.05	-0.76	-0.67	0.77
1	1	117	24	19	0.22	-0.06	2.30	-0.89	-0.89	-0.67	1.18
1	1	118	9	184	0.17	-0.98	0.34	-0.29	-0.05	-0.67	-0.08
1	1	119	11	43	0.12	-0.82	-0.89	-1.04	-0.79	-0.67	-1.65
1	1	120	9	257	0.29	2.17	-0.62	0.10	-0.05	-0.55	0.65
1	1	121	27	155	0.18	0.15	0.68	-0.84	-0.76	-0.67	0.81
1	1	122	20	215	0.11	0.33	-0.08	-0.69	-0.05	-0.67	-1.64
1	1	123	14	348	0.24	0.61	0.74	0.37	1.37	0.43	0.25
1	1	124	13	162	0.12	0.01	-1.01	-0.67	-0.05	-0.67	-1.57
1	1	125	7	366	0.22	1.62	0.35	-0.29	1.37	-0.67	-1.65
1	1	126	9	77	0.02	-0.88	-0.89	-0.78	-0.76	-0.67	0.66
1	1	127	25	253	0.14	0.70	0.68	0.16	-0.05	-0.67	0.61
1	1	128	9	84	0.09	-0.27	-1.27	-0.81	-0.76	-0.67	0.81
1	1	129	3	398	0.04	-0.03	-0.08	2.29	1.37	0.43	-1.98
1	1	130	9	412	0.05	0.90	-0.89	2.29	2.79	0.43	-0.48
1	1	131	8	309	0.06	1.43	-0.89	0.41	1.37	-0.67	0.46
1	1	132	8	267	0.08	0.10	0.68	0.37	-0.05	0.43	1.57
1	1	133	16	252	0.22	0.94	-0.08	-0.40	-0.14	0.43	0.43
1	1	134	15	119	0.13	-0.60	-0.99	-0.75	-0.76	0.43	0.24
1	1	135	13	48	0.09	-0.66	-0.92	-1.19	-1.11	-0.67	0.82
1	1	136	9	326	0.11	1.45	-0.08	0.32	1.37	-0.67	0.38
1	1	137	11	386	0.22	1.37	0.68	-0.66	-0.18	2.64	-1.36
1	1	138	20	255	0.15	0.43	-0.89	-0.34	-0.05	2.64	0.68
1	1	139	8	161	0.14	-0.68	-0.94	-0.08	-0.05	0.43	1.41
1	1	140	10	38	0.13	-1.22	-1.19	-0.92	-0.76	0.43	0.91
1	1	141	14	180	0.12	0.09	0.68	-0.56	-0.76	-0.67	0.07
1	1	142	13	427	0.21	2.01	0.62	2.29	2.79	0.43	-0.18
1	1	143	15	354	0.13	1.63	0.68	0.39	1.37	-0.67	0.26
1	1	144	11	301	0.14	0.82	-1.03	0.15	1.37	-0.67	-0.85
1	1	145	7	205	0.13	0.24	-0.08	-0.05	-0.05	-0.67	1.57
1	1	146	10	327	0.14	0.66	0.68	0.60	1.37	-0.67	0.42
1	1	147	11	403	0.11	0.92	2.30	1.22	1.37	0.43	-0.79
1	1	148	7	127	0.08	-0.20	-0.89	-0.10	-0.76	-0.67	0.69
1	1	149	6	217	0.11	0.41	-0.89	0.26	-0.05	-0.67	-0.34
1	1	150	11	323	0.1	0.96	-0.89	0.46	1.37	0.43	0.72
1	1	151	9	208	0.16	0.16	0.68	-0.60	-0.05	-0.67	0.43
1	1	152	14	405	0.14	-0.04	0.68	2.29	1.37	0.43	-2.24
1	1	153	10	98	0.08	-0.22	-0.89	-1.14	-0.76	-0.67	0.33
1	1	154	7	47	0.05	-0.99	-0.89	-1.16	-0.76	-0.67	1.03
1	1	155	8	182	0.15	-1.00	0.68	0.08	-0.76	-0.67	-0.46
1	1	156	9	153	0.07	-0.24	-0.93	0.04	-0.05	-0.67	1.57
1	1	157	23	65	0.08	-1.23	-0.89	-0.82	-0.76	-0.67	0.27
1	1	158	4	367	0.08	3.03	-0.08	0.15	-0.05	-0.67	-1.65
1	1	159	29	421	0.15	1.04	0.68	2.29	2.79	0.43	-0.84
1	1	160	10	111	0.14	-1.33	-0.89	0.06	-0.76	-0.67	-0.65
1	1	161	13	382	0.25	3.03	0.68	0.23	-0.05	-0.42	-1.68
1	1	162	10	142	0.1	-0.54	-0.89	-0.76	-0.05	-0.67	-0.73
1	1	163	14	433	0.32	1.31	-0.10	2.29	2.79	2.64	-0.81
1	1	164	14	50	0.08	-0.88	-0.08	-1.19	-1.11	-0.67	0.91
1	1	165	23	125	0.14	-0.78	0.68	-0.84	-0.76	-0.67	0.41
1	1	166	7	67	0.09	-0.95	0.68	-1.13	-0.96	-0.67	1.03
1	1	167	12	254	0.11	-0.53	0.68	0.02	-0.05	0.43	-0.24
1	1	168	11	113	0.17	-0.63	-0.08	-0.98	-0.92	-0.67	-0.30
1	1	169	9	83	0.04	-0.71	-0.89	-0.75	-0.76	-0.67	0.76
1	1	170	10	89	0.11	-1.18	-0.93	-0.19	-0.76	-0.67	0.66
1	1	171	8	44	0.06	-1.41	-0.89	-1.13	-0.76	-0.67	0.36
1	1	172	8	310	0.08	1.13	-0.99	0.45	1.37	-0.67	-0.08
1	1	173	8	6	0.08	-1.21	-0.89	-1.28	-1.11	-0.67	-1.64
1	1	174	14	295	0.12	-0.78	-0.08	0.38	-0.76	2.64	-1.00
1	1	175	12	316	0.19	1.00	-1.02	-0.29	1.37	-0.67	-1.61
1	1	176	2	346	0.15	0.38	-0.11	2.60	-0.05	0.43	0.69
1	1	177	12	194	0.17	-0.92	0.68	-0.46	-0.76	0.43	-0.16
1	1	178	17	396	0.19	1.51	2.30	0.45	1.37	-0.09	1.34
1	1	179	15	206	0.13	-1.30	-0.08	0.26	-0.76	0.43	-0.90
1	1	180	4	322	0.13	1.35	-0.89	-0.06	1.37	0.43	0.52
1	1	181	31	225	0.12	-0.25	0.68	0.14	-0.05	-0.67	0.21
1	1	182	5	185	0.14	0.46	-0.57	-0.29	-0.76	0.43	1.27
1	1	183	12	103	0.15	-0.68	-0.08	-0.63	-0.76	-0.67	1.39
1	1	184	2	151	0.06	-0.29	-0.89	0.06	-0.76	-0.67	-0.87
1	1	185	11	232	0.12	-0.39	0.68	0.22	-0.05	-0.67	-0.20
1	1	186	9	313	0.29	0.41	-0.89	1.32	1.37	-0.67	0.31
1	1	187	17	372	0.15	0.23	0.70	1.21	1.37	0.43	-0.74
1	1	188	9	407	0.09	0.41	2.30	1.70	1.37	0.43	-1.08
1	1	189	14	68	0.36	-0.83	0.19	-1.16	-1.09	0.43	0.95
1	1	190	9	37	0.08	-1.53	-1.27	-0.83	-0.76	-0.67	0.41
1	1	191	8	28	0.13	-1.33	-1.22	-1.10	-0.85	-0.67	-0.62
1	1	192	8	71	0.24	-1.65	-0.89	-0.19	-0.67	-0.67	0.43
1	1	193	9	7	0.2	-0.89	-0.89	-0.36	-0.84	2.64	0.50
1	1	194	16	53	0.05	-1.15	-0.89	-0.82	-0.76	-0.67	1.06
1	1	195	9	29	0.07	-0.60	-1.27	-1.08	-0.76	-0.67	-1.66
1	1	196	14	377	0.21	2.10	0.68	0.25	1.37	0.43	0.77
1	1	197	10	190	0.2	-0.45	-0.73	-0.74	-0.05	0.43	-0.77
1	1	198	11	15	0.13	-1.32	-1.03	-1.19	-1.11	0.43	0.88
1	1	199	13	344	0.23	0.11	0.68	0.14	-0.05	2.64	-0.39
1	1	200	12	128	0.1	0.05	-0.08	-0.81	-0.76	-0.67	0.90
1	1	201	17	36	0.06	-1.25	-1.27	-1.13	-0.76	-0.67	0.40
1	1	202	12	172	0.1	-0.33	-0.89	-0.44	-0.05	0.43	0.94
1	1	203	8	293	0.04	-0.74	1.11	0.50	-0.05	0.43	-1.22
1	1	204	4	165	0.06	0.02	-0.89	-0.67	-0.76	0.43	-1.69
1	1	205	13	187	0.13	0.43	-0.89	-0.71	-0.05	-0.67	-1.63
1	1	206	4	210	0.13	0.00	2.30	-0.78	-0.76	0.43	0.94
1	1	207	12	70	0.05	-0.72	-1.27	-0.81	-0.76	-0.67	0.69
1	1	208	13	156	0.14	0.14	-0.08	-0.21	-0.76	-0.67	0.93
1	1	209	10	317	0.17	0.87	2.30	0.21	-0.05	-0.67	1.31
1	1	210	19	197	0.23	-0.12	0.68	-0.79	-0.83	0.43	0.37
1	1	211	13	201	0.15	0.33	-0.98	-0.29	-0.05	-0.67	-0.85
1	1	212	12	94	0.15	-1.25	-0.08	-0.80	-0.76	-0.67	0.38
1	1	213	26	85	0.09	-0.35	-0.89	-0.98	-0.76	-0.67	-1.63
1	1	214	13	324	0.11	0.51	2.30	0.21	-0.05	0.43	0.66
1	1	215	9	95	0.08	-0.54	0.68	-1.18	-1.07	-0.67	1.01
1	1	216	23	126	0.14	-0.21	-0.08	-1.01	-0.77	-0.67	-1.64
1	1	217	10	256	0.2	0.88	0.68	-0.39	-0.05	-0.67	-0.79
1	1	218	19	356	0.34	-0.96	0.52	1.97	-0.20	-0.67	-2.18
1	1	219	14	147	0.11	0.17	-0.08	-0.77	-0.76	-0.67	0.27
1	1	220	14	240	0.2	0.44	0.86	0.03	-0.10	-0.67	1.57
1	1	221	13	191	0.18	0.14	-0.08	-0.75	-0.49	0.43	0.67
1	1	222	7	380	0.05	0.04	-0.08	1.70	1.37	0.43	-1.23
1	1	223	18	88	0.05	-0.46	-0.89	-0.77	-0.76	-0.67	1.00
1	1	224	12	417	0.1	1.69	0.68	2.29	2.79	-0.67	0.38
1	1	225	18	413	0.27	2.56	0.68	-0.04	1.37	2.64	0.59
1	1	226	9	401	0.08	0.81	-0.89	1.20	1.37	2.64	-0.61
1	1	227	8	49	0.08	-1.28	-0.89	-0.29	-0.76	-0.67	1.41
1	1	228	8	12	0.18	-1.32	-1.03	-0.93	-0.85	0.43	1.57
1	1	229	6	237	0.19	0.49	0.17	-0.73	-0.76	0.43	-1.52
1	1	230	16	266	0.12	-0.03	0.68	0.49	-0.05	0.43	0.54
1	1	231	5	82	0.12	-1.25	-1.12	-0.29	-0.05	-0.67	1.27
1	1	232	24	110	0.12	-0.42	-1.00	-0.71	-0.76	-0.67	-0.07
1	1	233	14	270	0.15	0.34	-1.11	0.45	1.37	-0.67	1.39
1	1	234	4	200	0.07	-0.30	-0.89	0.26	-0.05	0.43	1.57
1	1	235	16	263	0.29	0.70	0.35	-0.52	-0.18	0.43	-0.62
1	1	236	12	280	0.21	1.35	0.68	-0.41	-0.23	0.43	0.90
1	1	237	8	281	0.07	-0.96	0.68	0.48	-0.05	0.43	-1.23
1	1	238	7	287	0.13	0.20	0.68	0.45	-0.05	0.43	-0.55
1	1	239	16	116	0.18	-1.23	-0.08	-0.57	-0.76	-0.67	-0.37
1	1	240	31	321	0.42	-0.50	2.30	0.40	-0.12	-0.03	-1.00
1	1	241	16	420	0.13	1.51	-0.08	2.29	2.79	0.43	-0.49
1	1	242	8	199	0.12	0.05	-1.03	0.20	-0.05	-0.67	-0.07
1	1	243	8	314	0.07	0.99	-0.08	0.45	1.37	-0.67	0.72
1	1	244	7	289	0.06	-0.20	0.68	1.23	-0.05	0.43	0.35
1	1	245	15	216	0.16	-0.16	0.68	0.03	-0.05	-0.67	0.76
1	1	246	14	273	0.16	0.31	0.63	0.14	-0.05	0.43	0.07
1	1	247	20	132	0.18	-0.76	-0.08	-0.14	-0.72	-0.67	0.74
1	1	248	5	369	0.08	-0.74	0.68	1.70	-0.05	0.43	-2.35
1	1	249	11	432	0.33	0.40	1.35	2.13	1.37	2.64	-2.28
1	1	250	6	341	0.22	0.46	0.62	0.21	1.37	0.43	1.36
1	1	251	24	404	0.12	1.30	-0.89	2.29	2.79	-0.67	0.33
1	1	252	9	63	0.05	-1.23	-0.89	-0.75	-0.76	-0.67	0.65
1	1	253	11	226	0.16	0.32	-0.96	-0.44	-0.05	0.43	-0.89
1	1	254	11	133	0.09	-0.43	-0.89	-0.75	-0.05	-0.67	0.33
1	1	255	19	376	0.16	0.60	0.68	1.17	1.37	0.43	-0.70
1	1	256	8	80	0.06	-0.23	-0.89	-1.15	-0.76	-0.67	0.91
1	1	257	13	164	0.27	-0.58	0.15	-0.89	-0.87	0.43	-0.58
1	1	258	7	78	0.07	-0.82	-0.89	-0.29	-0.76	-0.67	1.33
1	1	259	5	422	0.05	1.83	-0.89	2.29	2.79	0.43	-0.23
1	1	260	10	23	0.11	-1.45	-1.23	-1.16	-0.83	-0.67	-0.10
1	1	261	8	60	0.05	-0.57	-1.27	-1.14	-0.76	-0.67	0.44
1	1	262	12	93	0.12	-0.88	-0.92	-0.78	-0.76	-0.67	-0.41
1	1	263	20	279	0.24	0.33	2.30	0.14	-0.19	-0.67	0.31
1	1	264	15	75	0.05	-0.67	-0.89	-1.10	-0.76	-0.67	0.39
1	1	265	15	351	0.04	0.27	-0.89	1.20	1.37	0.43	-0.45
1	1	266	7	408	0.25	2.71	-0.89	-0.27	1.37	2.64	0.06
1	1	267	10	57	0.07	-0.55	-1.27	-1.15	-0.76	-0.67	0.82
1	1	268	13	387	0.24	0.11	0.68	1.48	-0.05	2.64	-0.61
1	1	269	6	436	0.27	1.23	2.30	1.91	2.08	2.64	-1.04
1	1	270	11	202	0.09	0.20	-0.89	0.47	-0.05	-0.67	0.68
1	1	271	12	402	0.47	1.54	0.81	2.42	1.37	0.34	0.54
1	1	272	12	152	0.15	-0.05	-0.08	-0.83	-0.76	-0.67	-0.87
1	1	273	9	21	0.09	-1.48	-0.89	-1.15	-0.96	-0.67	0.95
1	1	274	11	69	0.05	-0.74	-0.89	-0.84	-0.76	-0.67	1.11
1	1	275	14	249	0.24	-0.06	-0.77	-0.01	-0.15	2.64	0.62
1	1	276	4	318	0.08	-1.02	0.68	1.35	-0.05	0.43	-1.21
1	1	277	19	247	0.15	0.09	0.68	-0.36	-0.05	0.43	0.65
1	1	278	14	297	0.17	1.23	0.68	0.23	-0.05	0.43	0.58
1	1	279	8	338	0.08	0.88	0.68	0.45	1.37	-0.67	1.41
1	1	280	13	171	0.1	-0.38	-0.92	0.00	-0.05	-0.67	0.06
1	1	281	14	371	0.28	-0.93	2.30	0.46	-0.20	-0.12	-2.31
1	1	282	24	173	0.33	-0.28	0.73	-0.75	-0.82	0.43	0.97
1	1	283	18	355	0.14	1.38	0.68	-0.21	1.37	0.43	0.42
1	1	284	4	345	0.11	0.83	-0.49	-0.66	1.37	0.43	-1.59
1	1	285	11	56	0.05	-0.83	-1.27	-0.78	-0.76	-0.67	1.01
1	1	286	7	370	0.19	1.55	0.68	-0.38	1.37	0.43	-1.02
1	1	287	10	64	0.06	-0.48	-0.89	-1.19	-1.11	-0.67	0.41
1	1	288	9	167	0.15	-0.36	-0.89	-0.71	-0.05	0.43	0.29
1	1	289	6	124	0.15	-1.28	-1.02	-0.29	-0.05	-0.67	-0.30
1	1	290	14	272	0.11	-0.70	0.68	0.43	-0.05	0.43	-0.78
1	1	291	25	188	0.09	0.21	-0.89	0.05	-0.05	-0.67	0.52
1	1	292	23	435	0.3	1.05	2.30	2.59	2.79	0.43	-1.56
1	1	293	8	8	0.08	-1.66	-1.27	-1.18	-0.94	-0.67	1.07
1	1	294	14	431	0.2	2.14	2.30	2.29	2.79	-0.60	0.40
1	1	295	3	96	0.08	-0.16	-0.89	-0.58	-0.76	-0.67	1.57
1	1	296	9	87	0.06	-0.46	-1.27	-0.78	-0.76	-0.67	0.39
1	1	297	6	414	0.04	0.90	-0.89	2.29	2.79	0.43	-0.85
1	1	298	10	230	0.18	0.31	-0.89	0.32	-0.12	0.43	0.71
1	1	299	13	58	0.11	-0.84	-0.95	-1.16	-0.98	-0.67	-0.08
1	1	300	11	4	0.31	0.63	-1.03	-0.85	-0.76	2.64	0.96
1	1	301	24	277	0.18	0.59	0.68	0.21	-0.08	0.43	0.76
1	1	302	13	130	0.1	-0.85	-0.08	-0.61	-0.76	-0.67	-0.80
1	1	303	6	131	0.17	-0.53	0.68	-0.61	-0.52	-0.67	1.57
1	1	304	10	55	0.09	-1.09	-1.27	-0.81	-0.76	-0.67	0.53
1	1	305	21	269	0.11	-1.16	0.68	0.48	-0.05	-0.67	-2.25
1	1	306	21	91	0.1	-0.45	-0.08	-1.19	-1.11	-0.67	0.57
1	1	307	9	104	0.08	-0.46	-0.98	-0.87	-0.76	-0.67	-1.12
1	1	308	19	32	0.07	-1.02	-0.89	-1.19	-1.11	-0.67	0.97
1	1	309	4	268	0.13	0.22	-0.89	0.07	-0.05	2.64	1.46
1	1	310	4	59	0.03	-0.93	-0.89	-1.14	-0.76	-0.67	0.69
1	1	311	4	2	0.12	-0.52	-0.89	-0.66	-0.76	2.64	1.22
1	1	312	8	3	0.15	-1.66	-1.22	-1.08	-0.89	-0.67	1.57
1	1	313	8	141	0.14	-0.61	-0.89	-0.21	-0.76	0.43	0.62
1	1	314	14	183	0.08	-0.35	-0.89	0.45	-0.05	-0.67	0.46
1	1	315	6	192	0.15	0.62	-0.89	-0.58	-0.76	0.43	-0.26
1	1	316	6	196	0.17	-0.81	-0.76	-0.07	-0.05	0.43	0.02
1	1	317	15	92	0.04	-0.75	-0.89	-0.75	-0.76	-0.67	0.36
1	1	318	6	214	0.21	1.06	0.17	-0.90	-0.64	-0.67	-1.66
1	1	319	9	389	0.05	0.14	0.68	1.70	1.37	0.43	-1.23
1	1	320	8	381	0.26	-0.31	2.30	1.86	-0.05	-0.12	-0.99
1	1	321	2	438	0.38	2.15	1.49	6.57	1.37	0.43	-0.28
1	1	322	18	224	0.09	-0.62	0.68	0.03	-0.05	-0.67	-0.65
1	1	323	7	361	0.05	0.63	-0.89	1.20	1.37	0.43	-0.72
1	1	324	8	352	0.12	0.50	-0.89	1.20	1.37	0.43	-0.08
1	1	325	8	234	0.1	-0.43	0.68	-0.12	-0.05	0.43	1.46
1	1	326	4	17	0.04	-1.04	-0.08	-1.19	-1.11	-0.67	1.57
1	1	327	3	5	0.04	-2.07	-0.89	-1.18	-1.11	-0.67	0.90
1	1	328	9	304	0.1	-0.22	0.68	1.23	-0.05	0.43	-0.46
1	1	329	9	350	0.09	-1.17	2.30	0.45	-0.76	-0.67	-2.26
1	1	330	12	294	0.1	0.98	-0.92	0.45	1.37	-0.67	0.42
1	1	331	9	307	0.16	0.42	-1.14	0.45	1.37	0.43	1.02
1	1	332	9	439	0.49	1.48	1.28	4.62	3.89	0.92	-2.46
1	1	333	10	148	0.16	-1.03	-0.57	-0.22	-0.76	0.43	-0.12
1	1	334	6	424	0.25	1.52	2.30	0.32	1.37	2.64	0.77
1	1	335	19	52	0.13	-1.61	-0.91	-0.75	-0.78	-0.67	-0.31
1	1	336	12	31	0.05	-1.28	-0.89	-1.19	-1.11	-0.67	0.60
1	1	337	19	123	0.2	-1.52	-0.08	-0.20	-0.76	-0.67	-0.94
1	1	338	11	218	0.17	-1.37	0.54	0.45	-0.76	-0.67	-2.18
1	1	339	18	262	0.2	0.88	0.68	-0.51	-0.17	0.43	0.51
1	1	340	3	325	0.07	-1.02	0.68	0.58	-0.05	0.43	-2.46
1	1	341	7	212	0.1	-1.39	0.68	-0.26	-0.76	0.43	-1.09
1	1	342	12	41	0.1	-1.75	-0.92	-0.62	-0.76	-0.67	-0.88
1	1	343	7	290	0.08	0.87	-1.27	0.46	1.37	-0.67	0.86
1	1	344	8	139	0.12	-1.50	-0.89	0.01	-0.76	0.43	-0.63
1	1	345	14	360	0.27	2.68	0.68	0.44	-0.25	0.43	0.22
1	1	346	6	181	0.05	-0.41	-0.89	-0.29	-0.05	0.43	0.55
1	1	347	13	353	0.11	1.54	0.68	0.47	1.37	-0.67	0.87
1	1	348	8	349	0.13	1.34	0.30	0.15	1.37	-0.67	-0.87
1	1	349	6	61	0.04	-0.65	-0.89	-1.14	-0.76	-0.67	0.94
1	1	350	21	336	0.25	0.45	0.68	0.04	-0.12	2.64	0.86
1	1	351	12	235	0.26	-0.42	0.36	-0.39	-0.76	2.64	0.12
1	1	352	10	115	0.16	-1.06	-1.08	-0.38	-0.05	-0.67	0.60
1	1	353	5	339	0.15	-0.28	0.22	2.29	-0.05	0.43	-0.55
1	1	354	16	397	0.34	-0.72	2.30	1.86	-0.14	-0.26	-2.30
1	1	355	4	236	0.15	-0.45	0.68	-0.08	-0.05	-0.67	-1.43
1	1	356	11	160	0.1	-0.31	-1.27	0.12	-0.05	-0.67	0.58
1	1	357	15	10	0.12	-1.28	-1.27	-1.05	-0.83	-0.67	-1.64
1	1	358	4	392	0.2	0.97	2.30	0.54	1.37	0.43	0.28
1	1	359	10	144	0.11	0.46	-0.89	-0.29	-0.76	-0.67	0.75
1	1	360	7	213	0.07	-0.32	-0.89	0.48	-0.05	0.43	0.59
1	1	361	9	394	0.28	2.75	0.68	-0.09	-0.29	2.64	0.44
1	1	362	7	340	0.2	0.97	-1.00	0.03	1.37	0.43	-0.83
1	1	363	21	251	0.24	1.15	0.68	-0.21	-0.12	-0.67	0.79
1	1	364	11	118	0.1	0.14	-0.89	-0.84	-0.76	-0.67	0.42
1	1	365	12	99	0.04	-0.35	-0.89	-0.79	-0.76	-0.67	0.76
1	1	366	8	388	0.15	1.15	2.30	0.73	1.37	-0.67	0.52
1	1	367	8	260	0.14	0.41	0.68	0.40	-0.05	-0.67	-0.36
1	1	368	12	117	0.09	-0.93	-0.89	-0.04	-0.76	-0.67	0.01
1	1	369	12	303	0.05	-0.52	0.68	1.23	-0.05	0.43	-0.83
1	1	370	7	395	0.21	1.06	2.30	0.16	-0.05	2.64	0.75
1	1	371	17	81	0.13	-1.49	-0.91	-0.30	-0.76	-0.67	-0.20
1	1	372	10	209	0.28	0.96	0.60	-0.64	-0.76	-0.67	-0.04
1	1	373	17	66	0.08	-1.33	-0.89	-0.81	-0.76	-0.67	-0.51
1	1	374	14	136	0.12	-0.28	-0.92	-0.50	-0.05	-0.67	1.05
1	1	375	20	198	0.09	-0.42	-0.89	0.45	-0.05	-0.67	-0.41
1	1	376	16	26	0.08	-1.40	-1.27	-1.15	-0.76	-0.67	0.81
1	1	377	7	343	0.22	0.87	-0.89	-0.65	-0.35	2.64	-1.45
1	1	378	10	39	0.06	-1.04	-1.27	-1.11	-0.76	-0.67	0.91
1	1	379	8	229	0.09	0.17	0.68	-0.63	-0.05	-0.67	-0.85
1	1	380	15	175	0.14	0.41	-0.89	-0.46	-0.05	-0.67	0.41
1	1	381	13	35	0.07	-1.28	-1.27	-0.82	-0.76	-0.67	1.03
1	1	382	14	220	0.09	-0.81	-0.08	0.45	-0.05	-0.67	-0.96
1	1	383	11	283	0.12	0.38	2.30	-0.05	-0.05	-0.67	1.46
1	1	384	10	134	0.12	-0.09	1.02	-0.78	-0.76	-0.67	1.36
1	1	385	14	416	0.08	1.33	-0.89	2.29	2.79	0.43	-0.30
1	1	386	15	76	0.18	-1.35	-0.94	-0.63	-0.81	0.43	0.43
1	1	387	23	157	0.08	-0.37	-0.89	-0.04	-0.05	-0.67	0.67
1	1	388	4	391	0.26	1.95	0.49	0.42	-0.05	2.64	0.27
1	1	389	6	105	0.08	-0.68	0.68	-1.25	-1.11	-0.67	-1.44
1	1	390	8	315	0.16	0.60	-1.08	0.37	1.37	0.43	0.25
1	1	391	14	72	0.04	-0.78	-0.89	-0.76	-0.76	-0.67	0.96
1	1	392	8	259	0.15	0.35	0.79	-0.12	-0.05	0.43	1.28
1	1	393	18	222	0.1	-0.38	-0.08	0.45	-0.05	-0.67	-0.53
1	1	394	11	333	0.06	1.49	-0.08	0.46	1.37	-0.67	0.89
1	1	395	15	276	0.1	-0.28	0.68	0.32	-0.05	0.43	-0.74
1	1	396	9	16	0.11	-1.53	-1.27	-1.19	-1.03	-0.67	0.53
1	1	397	9	364	0.08	0.27	-0.08	1.20	1.37	0.43	-0.75
1	1	398	10	211	0.14	0.00	-1.00	-0.05	-0.05	0.43	0.18
1	1	399	7	384	0.25	0.30	0.74	1.76	1.37	0.43	-0.80
1	1	400	6	177	0.14	1.15	-0.08	-0.61	-0.76	-0.67	0.67
1	1	401	19	203	0.11	-0.13	-1.01	0.05	-0.05	0.43	0.76
1	1	402	8	312	0.09	0.73	-0.08	0.45	1.37	-0.67	1.41
1	1	403	9	258	0.2	-0.18	-0.89	2.29	-0.05	-0.67	-0.38
1	1	404	11	244	0.16	-1.09	0.68	0.39	-0.76	0.43	-0.94
1	1	405	11	319	0.21	0.03	0.54	2.29	-0.05	-0.67	-0.36
1	1	406	15	159	0.08	-0.10	-1.27	0.08	-0.05	-0.67	0.91
1	1	407	33	221	0.14	0.64	-0.08	-0.05	-0.05	-0.67	0.68
1	1	408	11	112	0.09	-0.50	0.68	-1.18	-1.08	-0.67	0.48
1	1	409	17	399	0.31	1.42	0.73	0.26	1.37	2.64	0.75
1	1	410	5	20	0.08	-1.28	-1.27	-1.05	-0.90	-0.67	-1.12
1	1	411	6	34	0.08	-0.85	-0.08	-1.25	-1.11	-0.67	-1.44
1	1	412	6	426	0.2	3.36	0.68	-0.27	1.37	2.64	-0.22
1	1	413	12	285	0.1	0.72	-1.24	0.30	1.37	-0.67	0.32
1	1	414	8	363	0.08	1.28	1.11	0.45	1.37	-0.67	1.36
1	1	415	8	437	0.27	1.02	1.00	3.07	2.79	2.64	-2.30
1	1	416	7	284	0.19	0.20	-0.66	-0.74	-0.05	2.64	-0.71
1	1	417	9	264	0.06	0.29	-0.89	0.45	1.37	-0.67	0.66
1	1	418	21	238	0.21	0.24	-0.08	-0.09	-0.05	0.43	0.69
1	1	419	3	296	0.02	-0.47	0.68	0.03	-0.05	0.43	-2.34
1	1	420	10	100	0.16	-0.44	-0.97	-0.87	-0.76	0.43	1.21
1	1	421	11	174	0.15	-0.61	0.68	-0.02	-0.76	-0.67	0.17
1	1	422	9	311	0.25	1.52	-0.53	-0.55	-0.76	2.64	0.36
1	1	423	6	135	0.09	-0.58	-1.27	0.12	-0.05	-0.67	1.29
1	1	424	5	368	0.15	2.37	0.68	0.36	1.37	-0.67	0.68
1	1	425	9	373	0.12	1.65	0.68	0.39	1.37	0.43	-0.01
1	1	426	10	228	0.13	0.42	-0.08	-0.33	-0.05	-0.67	-0.92
1	1	427	20	86	0.22	-1.23	-0.93	-0.85	-0.83	0.43	-0.46
1	1	428	7	106	0.06	-0.07	-0.89	-0.76	-0.76	-0.67	0.98
1	1	429	8	138	0.12	-0.45	-0.99	-0.78	-0.76	0.43	-0.77
1	1	430	8	30	0.12	-0.51	1.05	-1.19	-1.11	-0.67	1.36
1	1	431	19	357	0.16	1.03	0.68	-0.27	-0.05	2.64	0.61
1	1	432	20	406	0.38	0.02	2.30	0.99	-0.05	2.64	-1.12
1	1	433	10	302	0.26	1.83	0.53	-0.09	-0.33	0.43	0.24
1	1	434	5	73	0.04	-0.72	-0.89	-0.95	-0.76	-0.67	0.82
1	1	435	1	440	0	5.51	0.68	3.07	4.20	2.64	0.50
1	1	436	14	300	0.29	2.04	0.68	0.38	-0.15	-0.67	0.69
1	1	437	19	74	0.1	-1.10	-0.89	-0.81	-0.76	-0.67	-0.90
1	1	438	7	347	0.13	1.21	-0.08	0.41	1.37	0.43	0.59
1	1	439	31	374	0.28	-0.27	0.87	0.49	-0.05	2.64	-1.24
1	1	440	17	233	0.2	-0.48	0.68	-0.10	-0.13	0.43	0.60

Now let us understand what each column in the above table means:

Segment.Level - Level of the cell. In this case, we have performed Vector Quantization for depth 1. Hence Segment Level is 1.
Segment.Parent - Parent segment of the cell.
Segment.Child (Cell.Number) - The children of a particular cell. In this case, it is the total number of cells at which we achieved the defined compression percentage.
n - No of points in each cell.
Cell.ID - Cell_ID’s are generated for the multivariate data using 1-D Sammon’s Projection algorithm.
Quant.Error - Quantization Error for each cell.

All the columns after this will contain centroids for each cell. They can also be called a codebook, which represents a collection of all centroids or codewords.

Now, let’s check the compression summary for HVT (map A). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapA_compression_summary <- map_A[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapA_compression_summary)

segmentLevel	noOfCells	noOfCellsBelowQuantizationError	percentOfCellsBelowQuantizationErrorThreshold	parameters
1	440	355	0.81	n_cells: 440 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 81% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

Now let’s try to understand plotHVT function. The parameters have been explained in detail below:

plotHVT(hvt.results, line.width, color.vec, pch1 = 21, centroid.size = 3, title = NULL, maxDepth = 1)

hvt.results - A list containing the output of the HVT function which has the details of the tessellations to be plotted.
line.width - A vector indicating the line widths of the tessellation boundaries for each layer.
color.vec - A vector indicating the colors of the tessellations boundaries at each layer.
pch1 - Symbol type of the centroids of the tessellations (parent levels). Refer points (default = 21)
centroid.size - Size of centroids of first level tessellations (default = 3).
title - Set a title for the plot (default = NULL).

Let’s plot the Voronoi tessellation for layer 1 (map A).

muHVT::plotHVT(map_A,
        line.width = c(0.4), 
        color.vec = c("#141B41"),
        centroid.size = 0.01,
        maxDepth = 1)

Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 440 cells in the dataset ’computers’

Heat Maps

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data.

Let’s have a look at the function hvtHmap that we will use to overlay features as heatmap.

hvtHmap(hvt.results, dataset, child.level, hmap.cols, color.vec ,line.width, palette.color = 6)

hvt.results - A list of results obtained from the HVT function (map_A).
dataset - A dataframe containing the variables to overlay as a heatmap. The user can pass an external dataset or the dataset that was used to perform hierarchical vector quantization. The dataset should have the same number of points as the dataset used to perform hierarchical Vector Quantization in the HVT function.
child.level - A number indicating the level for which the heat map is to be plotted.
hmap.cols - The column number of column name from the dataset indicating the variables for which the heat map is to be plotted. To plot the quantization error as heatmap, pass 'quant_error'. Similarly to plot the no of points in each cell as heatmap, pass 'no_of_points' as a parameter.
color.vec - A color vector such that length(color.vec) = child.level (default = NULL).
line.width - A line width vector such that length(line.width) = child.level (default = NULL).
palette.color - A number indicating the heat map color palette. 1 - rainbow, 2 - heat.colors, 3 - terrain.colors, 4 - topo.colors, 5 - cm.colors, 6 - BlCyGrYlRd (Blue,Cyan,Green,Yellow,Red) color (default = 6).
show.points - A boolean indicating whether the centroids should be plotted on the tessellations (default = FALSE).

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (n,price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "n",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
)

Figure 3: The Voronoi Tessellation with the heat map overlaid for No. of entities in each cell


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "price",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
)

Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’price’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "speed",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
)

Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’speed’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "hd",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
)

Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’hd’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ram",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
)

Figure 7: The Voronoi Tessellation with the heat map overlaid for variable ’ram’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "screen",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
)

Figure 8: The Voronoi Tessellation with the heat map overlaid for variable ’screen’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ads",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
)

Figure 9: The Voronoi Tessellation with the heat map overlaid for variable ’ads’ in the ’computers’ dataset

4 Map B : Compressed Novelty Map

Let us try to visualize the Map B from the flow diagram below.

Figure 10: Flow map with highlighted bounding box in red around map B

In this section, we will manually figure out the novelty cells from the plotted map A and store it in identified_Novelty_cells variable.

Note: For manual selecting the novelty cells from map A, one can enhance its interactivity by adding plotly elements to the code. This will transform map A into an interactive plot, allowing users to actively engage with the data. By hovering over the centroids of the cells, a tag containing segment child information will be displayed. Users can explore the map by hovering over different cells and selectively choose the novelty cells they wish to consider.

The removeNovelty function removes the identified novelty cell(s) from the dataset and stores those records separately.

It takes input as the cell number (Segment.Child) of the manually identified novelty cell(s) from the above table and the compressed HVT map (map A). It returns a list of two items: dataset with novelty records, and a subset of the dataset without the novelty records.

identified_Novelty_cells <<- c(73,321,332,338,435)
output_list <- removeNovelty(identified_Novelty_cells, map_A)

[1] “The following cell(s) have been removed as outliers from the dataset: 73 321 332 338 435”

data_with_novelty <- output_list[[1]]
dataset_without_novelty <- output_list[[2]]

Let’s have a look at the data with novelties.For the sake of brevity, we will only show the first 10 rows.

novelty_data <- data_with_novelty
novelty_data$Row.No <- row.names(novelty_data)
novelty_data <- novelty_data %>% dplyr::select("Row.No","Cell.ID","Cell.Number","price","speed","hd","ram","screen","ads")
colnames(novelty_data) <- c("Row.No","Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
novelty_data %>% head(100) %>% 
  as.data.frame() %>%
  Table(scroll = T, limit = 20)

Row.No	Cell.ID	Segment.Child	price	speed	hd	ram	screen	ads
1	429	73	3.0762240	0.6794579	0.0421969	4.2031619	0.4307274	0.7120258
2	438	321	2.6449847	0.6794579	6.5710368	1.3676416	0.4307274	0.6851382
3	438	321	1.6578103	2.2969425	6.5710368	1.3676416	0.4307274	-1.2507722
4	439	332	0.9130997	0.6794579	4.6232922	2.7854017	0.4307274	-2.4607161
5	439	332	1.8569770	1.1076156	4.6232922	4.2031619	2.6404120	-2.4607161
6	439	332	1.5192595	2.2969425	4.6232922	4.2031619	0.4307274	-2.4607161
7	439	332	1.4482522	1.1076156	4.6232922	4.2031619	0.4307274	-2.4607161
8	439	332	1.8569770	1.1076156	4.6232922	4.2031619	2.6404120	-2.4607161
9	439	332	1.1555636	2.2969425	4.6232922	2.7854017	0.4307274	-2.4607161
10	439	332	1.7773103	1.1076156	4.6232922	4.2031619	0.4307274	-2.4607161
11	439	332	1.3460710	0.6794579	4.6232922	4.2031619	0.4307274	-2.4607161
12	439	332	1.4482522	1.1076156	4.6232922	4.2031619	0.4307274	-2.4607161
13	218	338	-1.4959522	0.6794579	0.4473277	-0.7589986	-0.6741149	-2.2993903
14	218	338	-1.5115391	0.6794579	0.4473277	-0.7589986	-0.6741149	-1.9767386
15	218	338	-1.1668940	0.6794579	0.4473277	-0.7589986	-0.6741149	-2.2859465
16	218	338	-1.4959522	0.6794579	0.4473277	-0.7589986	-0.6741149	-2.4472723
17	218	338	-1.5981334	0.6794579	0.4473277	-0.7589986	-0.6741149	-2.2859465
18	218	338	-1.1668940	0.6794579	0.4473277	-0.7589986	-0.6741149	-2.2993903
19	218	338	-1.1668940	0.6794579	0.4473277	-0.7589986	-0.6741149	-2.4472723
20	218	338	-1.3227637	0.6794579	0.4473277	-0.7589986	-0.6741149	-1.9767386

4.1 Voronoi Tessellation with highlighted novelty cell

The plotCells function is used to plot the Voronoi tessellation using the compressed HVT map (map A) and highlights the identified novelty cell(s) in red on the map.

plotCells(identified_Novelty_cells, map_A,line.width = c(0.4),centroid.size = 0.01)

Figure 11: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

We pass the dataframe with novelty records to HVT function along with other model parameters mentioned below to generate map B (layer2)

Model Parameters

Number of Cells at each Level = 5
Maximum Depth = 1
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

colnames(data_with_novelty) <- c("Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
dataset_with_novelty <- data_with_novelty[,-1:-2]
map_B <- list()
mapA_scale_summary = map_A[[3]]$scale_summary
map_B <- muHVT::HVT(dataset_with_novelty,
                  n_cells = 5,
                  depth = 1,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F
                  )

The datatable displayed below is the summary from map B (layer 2).

summaryTable(map_B[[3]]$summary,scroll = T,limit = 500)

Segment.Level	Segment.Parent	Segment.Child	n	Cell.ID	Quant.Error	price	speed	hd	ram	screen	ads
1	1	1	2	5	0.01	-1.55	-0.08	0.45	-0.76	-0.67	-1.98
1	1	2	9	3	0.49	1.48	1.28	4.62	3.89	0.92	-2.46
1	1	3	2	2	0.38	2.15	1.49	6.57	1.37	0.43	-0.28
1	1	4	9	4	0.08	-1.33	0.68	0.45	-0.76	-0.67	-2.22
1	1	5	2	1	0.66	4.29	0.68	1.55	4.20	1.54	0.60

5 Map C : Compressed Map without Novelty

Let us try to visualize the compressed Map C from the flow diagram below.

Figure 12:Flow map with highlighted bounding box in red around compressed map C

5.0.1 Iteration 1:

With the Novelties removed, we construct another hierarchical Voronoi tessellation map C layer 2 on the dataset without Novelty and below mentioned model parameters.

Model Parameters

Number of Cells at each Level = 10
Maximum Depth = 2
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

map_C <- list()
mapA_scale_summary = map_A[[3]]$scale_summary
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 10,
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F,
                  scale_summary = mapA_scale_summary)

Now let’s check the compression summary for HVT (map C). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)

segmentLevel	noOfCells	noOfCellsBelowQuantizationError	percentOfCellsBelowQuantizationErrorThreshold	parameters
1	10	0	0	n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2	100	7	0.07	n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 7% of the cells have hit the quantization threshold error in level 2

5.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression at depth 2. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

Number of Cells at each Level = 23
Maximum Depth = 2
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

map_C <- list()
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 23,    #23
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F,
                  scale_summary = mapA_scale_summary)

The datatable displayed below is the summary from map C (layer2).

summaryTable(map_C[[3]]$summary,scroll = T,limit = 500)

Segment.Level	Segment.Parent	Segment.Child	n	Cell.ID	Quant.Error	price	speed	hd	ram	screen	ads
1	1	1	391	512	0.56	-1.27	-1.04	-0.91	-0.83	-0.59	0.86
1	1	2	105	96	1.02	2.08	0.75	0.11	0.71	2.64	0.56
1	1	3	190	55	1.37	1.48	0.02	2.32	2.63	0.12	-0.22
1	1	4	86	15	0.94	0.87	1.27	2.61	2.57	1.33	-1.76
1	1	5	257	326	0.49	0.41	0.45	-0.08	-0.20	-0.67	0.66
1	1	6	116	201	0.77	1.57	0.45	-0.24	0.27	-0.30	-1.30
1	1	7	235	228	0.58	0.85	-0.82	0.42	1.37	-0.47	0.52
1	1	8	149	185	0.84	-0.79	1.73	0.76	-0.22	-0.28	-1.75
1	1	9	353	306	0.69	-0.77	0.55	0.28	-0.22	-0.17	-0.79
1	1	10	234	297	0.54	0.25	0.50	-0.11	-0.16	0.43	0.66
1	1	11	330	478	0.56	-1.18	-0.79	-0.70	-0.76	-0.52	-0.69
1	1	12	373	434	0.6	-0.50	0.40	-0.84	-0.84	-0.56	0.66
1	1	13	209	147	0.81	1.32	0.99	0.36	1.37	-0.01	0.70
1	1	14	337	471	0.36	-0.51	-0.97	-0.87	-0.75	-0.67	0.57
1	1	15	283	404	0.71	0.06	-0.40	-0.78	-0.40	-0.42	-1.42
1	1	16	168	142	0.85	0.13	0.66	0.48	-0.02	2.64	-0.97
1	1	17	285	384	0.53	-0.04	-0.96	0.05	-0.12	-0.65	0.54
1	1	18	118	301	0.75	0.35	2.30	-0.13	-0.29	-0.28	0.82
1	1	19	256	414	0.47	-0.47	-0.84	-0.43	-0.48	0.43	0.60
1	1	20	78	57	1	0.18	1.67	2.05	1.31	0.39	-1.97
1	1	21	171	360	0.64	0.24	-0.46	-0.37	-0.41	2.64	0.59
1	1	22	170	130	1.28	0.28	-0.01	1.64	0.99	0.21	-0.62
1	1	23	89	219	0.88	1.98	0.59	0.20	-0.20	0.12	0.58
2	1	1	19	528	0.09	-1.31	-1.17	-1.19	-1.11	-0.67	0.99
2	1	2	18	479	0.31	-1.42	-0.87	-0.21	-0.72	-0.67	0.53
2	1	3	28	521	0.1	-1.47	-1.22	-1.11	-0.76	-0.67	0.89
2	1	4	10	530	0.14	-1.84	-1.04	-1.20	-1.11	-0.67	0.79
2	1	5	27	519	0.1	-1.20	-1.09	-1.19	-1.11	-0.67	0.50
2	1	6	13	524	0.18	-1.36	-1.10	-0.99	-0.87	0.43	1.36
2	1	7	5	523	0.04	-1.19	-1.27	-0.78	-0.76	-0.67	1.57
2	1	8	23	494	0.06	-1.18	-0.89	-0.78	-0.76	-0.67	0.72
2	1	9	27	514	0.14	-1.35	-1.27	-1.08	-0.76	-0.67	0.39
2	1	10	35	510	0.1	-1.08	-1.27	-0.88	-0.76	-0.67	0.90
2	1	11	6	529	0.15	-1.22	-0.62	-1.19	-1.11	-0.67	1.50
2	1	12	10	513	0.07	-0.95	-0.89	-0.85	-0.76	-0.67	1.57
2	1	13	10	517	0.22	-1.70	-1.19	-0.73	-0.83	-0.67	0.57
2	1	14	11	472	0.17	-1.19	-1.13	-0.33	-0.05	-0.67	0.97
2	1	15	11	507	0.19	-1.00	-0.93	-1.17	-0.63	-0.67	0.97
2	1	16	7	531	0.07	-1.58	-1.27	-1.18	-0.96	-0.67	1.57
2	1	17	26	499	0.09	-1.31	-0.89	-0.92	-0.76	-0.67	0.36
2	1	18	23	515	0.07	-0.99	-0.89	-1.19	-1.11	-0.67	0.92
2	1	19	13	520	0.14	-1.56	-0.98	-0.74	-0.76	-0.67	1.34
2	1	20	15	489	0.12	-1.05	-0.89	-0.26	-0.76	-0.67	1.31
2	1	21	16	503	0.18	-1.48	-0.79	-0.73	-0.76	-0.67	0.76
2	1	22	15	518	0.26	-1.37	-1.07	-1.09	-1.07	0.43	0.66
2	1	23	23	500	0.07	-1.03	-0.89	-0.82	-0.76	-0.67	1.06
2	2	1	4	108	0.12	2.31	0.68	-0.12	-0.05	2.64	1.31
2	2	2	3	118	0.11	2.53	0.68	-0.47	-0.76	2.64	0.36
2	2	3	7	99	0.21	1.06	0.80	0.29	1.37	2.64	0.35
2	2	4	2	89	0.06	1.09	2.30	0.25	-0.05	2.64	1.57
2	2	5	7	59	0.13	3.21	0.68	-0.27	1.37	2.64	0.53
2	2	6	7	88	0.11	1.85	0.68	0.16	1.37	2.64	0.85
2	2	7	2	145	0.11	1.14	0.68	0.10	-0.05	2.64	1.36
2	2	8	3	38	0.11	3.32	0.68	-0.27	1.37	2.64	-0.78
2	2	9	6	100	0.17	2.86	0.68	0.10	-0.05	2.64	0.48
2	2	10	2	24	0.11	3.50	2.30	1.23	-0.05	2.64	-0.23
2	2	11	2	12	0.38	4.64	0.68	1.36	0.66	2.64	0.37
2	2	12	6	56	0.25	1.52	2.30	0.32	1.37	2.64	0.77
2	2	13	4	50	0.34	1.64	0.49	1.90	1.37	2.64	0.14
2	2	14	13	76	0.22	2.42	0.68	0.03	1.37	2.64	0.53
2	2	15	4	103	0.12	1.08	2.30	0.14	-0.05	2.64	0.52
2	2	16	11	167	0.13	1.23	0.68	-0.20	-0.05	2.64	0.59
2	2	17	1	107	0	1.99	-0.08	0.49	-0.05	2.64	-0.62
2	2	18	4	84	0.15	1.52	0.68	0.34	1.37	2.64	1.36
2	2	19	5	169	0.17	1.81	-0.08	-0.29	-0.19	2.64	0.99
2	2	20	4	70	0.11	2.69	-0.89	-0.27	1.37	2.64	0.69
2	2	21	4	122	0.11	1.99	0.68	0.22	-0.05	2.64	0.57
2	2	22	3	60	0.12	2.74	-0.89	-0.27	1.37	2.64	-0.78
2	2	23	1	195	0	1.63	0.68	-0.95	-0.76	2.64	0.69
2	3	1	5	39	0.04	1.54	-0.08	2.29	2.79	0.43	-0.81
2	3	2	3	9	0.05	1.71	-0.89	2.29	2.79	2.64	-0.55
2	3	3	10	51	0.07	1.10	-0.08	2.29	2.79	0.43	-0.56
2	3	4	5	32	0.39	3.74	-0.10	2.60	1.37	-0.67	0.96
2	3	5	3	19	0.02	2.16	2.30	2.29	2.79	-0.67	0.04
2	3	6	1	20	0	2.03	2.30	2.29	2.79	0.43	0.35
2	3	7	16	49	0.18	1.49	-0.89	2.29	2.79	0.43	-0.06
2	3	8	7	40	0.09	1.80	-0.08	2.29	2.79	0.43	-0.26
2	3	9	2	54	0.05	2.90	0.68	3.73	-0.05	-0.67	0.69
2	3	10	1	91	0	1.16	0.68	2.29	1.37	-0.67	0.04
2	3	11	7	46	0.05	1.16	0.68	2.29	2.79	0.43	-0.49
2	3	12	4	77	0.12	1.66	0.68	2.44	1.37	0.43	0.80
2	3	13	7	37	0.08	1.63	0.68	2.29	2.79	0.43	-0.73
2	3	14	12	53	0.1	1.69	0.68	2.29	2.79	-0.67	0.38
2	3	15	2	43	0.04	3.07	0.68	2.29	1.37	-0.67	-0.87
2	3	16	5	23	0.04	2.06	2.30	2.29	2.79	-0.67	0.35
2	3	17	12	45	0.06	0.92	0.68	2.29	2.79	0.43	-0.86
2	3	18	5	82	0.13	1.35	0.68	2.47	1.37	0.43	0.11
2	3	19	5	17	0.04	2.23	2.30	2.29	2.79	-0.67	0.69
2	3	20	24	61	0.12	1.30	-0.89	2.29	2.79	-0.67	0.33
2	3	21	9	36	0.18	2.07	0.68	2.29	2.79	0.43	-0.03
2	3	22	13	52	0.11	0.65	-0.08	2.29	2.79	0.43	-1.03
2	3	23	32	48	0.13	1.15	-0.89	2.29	2.79	0.43	-0.66
2	4	1	5	7	0.11	0.51	2.30	2.05	1.37	2.64	-2.33
2	4	2	8	18	0.07	0.52	0.68	3.07	2.79	0.43	-2.36
2	4	3	6	41	0.05	0.79	0.68	2.29	2.79	0.43	-1.24
2	4	4	2	3	0	1.08	2.30	3.07	2.79	0.43	-2.46
2	4	5	3	8	0.13	1.91	0.43	2.29	2.79	2.64	-0.55
2	4	6	1	11	0	0.01	1.11	3.07	2.79	0.43	-2.45
2	4	7	3	26	0.03	0.73	-0.08	3.07	2.79	0.43	-1.98
2	4	8	1	4	0	0.99	-0.08	3.07	2.79	2.64	-1.98
2	4	9	3	2	0.05	1.48	2.30	2.29	2.79	2.64	-1.08
2	4	10	2	14	0.02	0.81	0.68	3.07	2.79	0.43	-2.46
2	4	11	1	6	0	0.74	2.30	3.07	2.79	0.43	-1.98
2	4	12	2	16	0.02	0.89	1.11	3.07	2.79	0.43	-1.98
2	4	13	2	29	0.04	0.84	2.30	1.70	1.37	2.64	-1.02
2	4	14	6	5	0.03	0.74	2.30	3.07	2.79	0.43	-2.40
2	4	15	7	1	0.23	1.03	1.16	3.07	2.79	2.64	-2.34
2	4	16	5	25	0.02	1.02	2.30	2.29	2.79	0.43	-1.23
2	4	17	6	22	0.25	0.85	-0.22	2.29	2.79	2.64	-0.97
2	4	18	2	13	0.05	1.21	0.68	2.29	2.79	2.64	-1.10
2	4	19	5	30	0.11	0.32	0.68	2.17	1.37	2.64	-2.29
2	4	20	6	10	0.04	0.69	1.11	3.07	2.79	0.43	-2.32
2	4	21	1	34	0	0.30	-0.08	2.29	1.37	2.64	-1.98
2	4	22	6	21	0.06	1.46	2.30	2.29	2.79	0.43	-1.01
2	4	23	3	28	0.01	1.00	2.30	2.29	2.79	0.43	-0.79
2	5	1	24	300	0.13	0.57	0.68	0.13	-0.05	-0.67	0.72
2	5	2	9	332	0.08	-0.44	0.68	-0.04	-0.05	-0.67	0.21
2	5	3	5	310	0.12	0.37	-0.08	0.29	-0.05	-0.67	-0.14
2	5	4	21	312	0.17	0.35	0.86	0.11	-0.05	-0.67	1.39
2	5	5	6	311	0.06	-0.28	0.68	0.45	-0.05	-0.67	0.30
2	5	6	8	287	0.15	0.45	0.68	0.16	-0.05	-0.67	-0.20
2	5	7	5	305	0.06	-0.07	0.68	0.45	-0.05	-0.67	0.48
2	5	8	7	333	0.12	0.31	0.68	-0.57	-0.05	-0.67	0.57
2	5	9	12	319	0.06	-0.12	0.68	0.04	-0.05	-0.67	0.14
2	5	10	13	358	0.15	-0.04	-0.08	0.00	-0.05	-0.67	0.83
2	5	11	11	371	0.16	0.41	0.68	-0.40	-0.76	-0.67	0.94
2	5	12	7	314	0.16	1.02	0.68	-0.39	-0.15	-0.67	1.12
2	5	13	17	363	0.2	0.66	0.63	-0.69	-0.76	-0.67	0.15
2	5	14	15	278	0.11	1.06	0.68	0.06	-0.05	-0.67	0.76
2	5	15	8	356	0.16	0.13	0.79	-0.13	-0.76	-0.67	0.38
2	5	16	11	275	0.19	1.06	0.68	-0.09	-0.05	-0.67	0.19
2	5	17	23	334	0.13	0.51	-0.08	0.01	-0.05	-0.67	0.91
2	5	18	17	324	0.11	0.69	-0.08	-0.08	-0.05	-0.67	0.46
2	5	19	12	402	0.15	0.15	-0.08	-0.20	-0.76	-0.67	0.94
2	5	20	7	386	0.14	1.08	-0.08	-0.62	-0.76	-0.67	0.63
2	5	21	12	336	0.2	-0.17	0.68	-0.04	-0.05	-0.67	0.86
2	5	22	7	364	0.13	0.24	-0.08	-0.05	-0.05	-0.67	1.57
2	5	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	6	1	6	184	0.1	2.46	0.68	0.21	-0.05	-0.67	-0.87
2	6	2	9	133	0.29	1.43	0.51	-0.44	1.37	0.43	-1.15
2	6	3	2	233	0.08	1.08	0.68	0.47	-0.05	-0.67	-0.71
2	6	4	9	247	0.22	0.90	0.59	-0.69	-0.13	0.43	-1.39
2	6	5	2	194	0.08	2.30	0.30	-0.32	-0.05	-0.67	-1.71
2	6	6	8	238	0.26	1.04	0.58	-0.29	-0.14	0.43	-0.77
2	6	7	9	127	0.06	2.98	0.68	0.28	-0.05	-0.67	-1.67
2	6	8	4	165	0.11	1.26	-0.08	-0.07	1.37	-0.67	-1.37
2	6	9	5	124	0.19	1.76	0.53	-0.29	1.37	-0.67	-1.66
2	6	10	1	101	0	3.77	0.68	0.15	-0.05	-0.67	-1.72
2	6	11	5	292	0.22	1.44	0.53	-0.77	-0.33	-0.67	-1.67
2	6	12	1	365	0	1.35	-0.08	-0.67	-0.76	-0.67	-0.62
2	6	13	4	149	0.06	1.43	0.68	0.15	1.37	-0.67	-0.87
2	6	14	10	197	0.2	1.57	0.68	-0.47	-0.33	0.43	-1.64
2	6	15	3	159	0.2	1.48	2.30	0.28	-0.05	-0.67	-0.87
2	6	16	3	280	0.1	0.85	-0.08	0.22	-0.05	-0.67	-0.68
2	6	17	9	191	0.15	1.09	-0.93	-0.29	1.37	-0.67	-1.62
2	6	18	4	140	0.08	3.03	-0.08	0.15	-0.05	-0.67	-1.65
2	6	19	7	294	0.05	0.73	0.68	-0.69	-0.05	-0.67	-1.60
2	6	20	3	112	0.09	2.96	0.68	0.12	-0.05	0.43	-1.72
2	6	21	2	270	0.04	1.61	0.68	-0.71	-0.05	-0.67	-0.87
2	6	22	8	279	0.09	0.84	0.68	-0.36	-0.05	-0.67	-0.84
2	6	23	2	237	0.08	2.30	-0.49	0.15	-0.05	-0.67	-1.12
2	7	1	8	210	0.16	0.51	-1.18	0.46	1.37	0.43	1.10
2	7	2	16	231	0.12	1.01	-0.91	0.35	1.37	-0.67	0.40
2	7	3	12	211	0.1	1.39	-0.89	0.37	1.37	-0.67	0.29
2	7	4	8	209	0.06	0.35	-0.89	1.20	1.37	-0.67	0.35
2	7	5	5	263	0.05	0.58	-0.89	0.45	1.37	-0.67	1.57
2	7	6	12	202	0.18	0.49	-1.02	0.55	1.37	0.43	0.36
2	7	7	14	264	0.08	0.43	-0.89	0.45	1.37	-0.67	0.64
2	7	8	2	166	0.09	0.99	-0.08	0.45	1.37	0.43	0.26
2	7	9	11	216	0.14	0.82	-1.03	0.15	1.37	-0.67	-0.85
2	7	10	20	220	0.09	1.20	-0.91	0.45	1.37	-0.67	0.84
2	7	11	9	265	0.07	0.48	-1.27	0.45	1.37	-0.67	0.66
2	7	12	14	190	0.21	1.09	-0.89	0.28	1.37	0.43	0.63
2	7	13	9	222	0.09	0.82	-1.06	0.45	1.37	-0.67	-0.07
2	7	14	7	168	0.2	0.97	-1.00	0.03	1.37	0.43	-0.83
2	7	15	9	204	0.2	0.72	-0.08	0.45	1.37	-0.55	1.43
2	7	16	17	243	0.09	0.82	-0.98	0.46	1.37	-0.67	0.95
2	7	17	7	262	0.04	0.22	-0.89	0.45	1.37	-0.67	0.04
2	7	18	7	285	0.07	0.28	-1.22	0.45	1.37	-0.67	1.12
2	7	19	6	179	0.12	1.31	-0.08	0.35	1.37	-0.67	-0.26
2	7	20	17	189	0.14	1.26	-0.08	0.46	1.37	-0.67	0.84
2	7	21	3	316	0.02	0.10	-1.27	0.45	1.37	-0.67	1.57
2	7	22	11	183	0.11	1.44	-0.08	0.34	1.37	-0.67	0.42
2	7	23	11	251	0.09	0.72	-1.24	0.29	1.37	-0.67	0.36
2	8	1	5	116	0.08	-0.74	0.68	1.70	-0.05	0.43	-2.35
2	8	2	2	282	0	-1.43	0.68	0.45	-0.76	0.43	-2.29
2	8	3	8	182	0.09	-0.47	2.30	0.49	-0.05	0.43	-0.85
2	8	4	10	236	0.19	-0.61	2.30	0.49	-0.05	-0.67	-1.01
2	8	5	13	87	0.15	-0.66	2.30	1.70	-0.05	-0.16	-2.28
2	8	6	4	117	0.13	-1.29	0.49	2.29	-0.76	-0.67	-2.17
2	8	7	2	64	0.02	-1.12	2.30	2.29	-0.76	-0.67	-2.37
2	8	8	3	104	0.06	-0.17	2.30	2.29	-0.05	-0.67	-0.98
2	8	9	4	170	0.08	-0.90	1.11	1.35	-0.05	0.43	-1.21
2	8	10	7	146	0.07	-0.91	2.30	0.47	-0.05	-0.67	-2.34
2	8	11	12	409	0.11	-0.78	2.30	-0.33	-0.76	-0.67	-1.03
2	8	12	7	241	0.1	-0.19	2.30	0.33	-0.05	-0.67	-0.89
2	8	13	4	258	0.09	0.04	2.30	-0.08	-0.05	-0.67	-1.04
2	8	14	9	330	0.09	-1.17	2.30	0.45	-0.76	-0.67	-2.26
2	8	15	6	212	0.13	-0.74	0.68	0.30	-0.05	0.43	-2.40
2	8	16	7	175	0.07	-0.53	2.30	0.49	-0.05	0.43	-1.23
2	8	17	4	109	0.04	-0.38	2.30	1.70	-0.05	0.43	-1.05
2	8	18	10	144	0.09	-0.89	0.68	1.70	-0.05	-0.67	-2.20
2	8	19	1	283	0	-1.59	0.68	-0.19	-0.05	0.43	-2.29
2	8	20	7	123	0.17	-0.94	2.30	0.46	-0.35	0.43	-2.27
2	8	21	3	329	0.06	-0.86	2.30	-0.29	-0.76	0.43	-0.98
2	8	22	21	277	0.11	-1.16	0.68	0.48	-0.05	-0.67	-2.25
2	8	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	9	1	33	331	0.22	-0.57	-0.08	0.41	-0.05	-0.67	-0.71
2	9	2	10	368	0.19	-0.75	0.68	-0.60	-0.76	0.43	-0.42
2	9	3	13	423	0.19	-1.02	0.68	-0.69	-0.70	-0.67	-0.83
2	9	4	8	335	0.2	-0.08	0.68	-0.49	-0.23	-0.67	-0.86
2	9	5	15	377	0.21	-1.31	-0.08	0.30	-0.76	0.43	-0.98
2	9	6	8	224	0.04	-0.74	1.11	0.50	-0.05	0.43	-1.22
2	9	7	22	261	0.13	-0.22	0.68	0.29	-0.05	0.43	-0.60
2	9	8	14	381	0.28	-0.89	0.57	-0.09	-0.51	-0.67	-0.23
2	9	9	14	257	0.08	-0.64	0.68	0.49	-0.05	0.43	-0.76
2	9	10	13	354	0.27	-1.37	0.33	0.47	-0.27	-0.67	-0.93
2	9	11	9	248	0.16	-1.00	0.68	0.57	-0.05	0.43	-1.22
2	9	12	23	323	0.13	-0.51	0.68	0.02	-0.05	-0.67	-0.46
2	9	13	10	344	0.25	-1.39	0.68	-0.30	-0.48	0.43	-1.23
2	9	14	11	267	0.11	-1.03	0.68	0.41	-0.05	0.43	-0.85
2	9	15	20	422	0.17	-1.43	0.68	-0.23	-0.72	-0.67	-1.02
2	9	16	23	206	0.26	-0.52	0.70	1.25	-0.05	0.43	-0.79
2	9	17	8	304	0.13	-0.86	0.68	-0.12	-0.05	0.43	-0.39
2	9	18	8	240	0.05	-0.72	1.11	0.50	-0.05	0.43	-0.83
2	9	19	32	303	0.35	-0.82	0.68	0.33	0.04	-0.67	-1.07
2	9	20	13	317	0.16	-1.11	0.68	0.34	-0.76	0.43	-0.90
2	9	21	10	308	0.22	-0.75	-0.24	0.33	-0.05	0.43	-0.69
2	9	22	33	296	0.2	-0.39	0.68	0.55	-0.05	-0.67	-0.56
2	9	23	3	196	0.03	-0.83	-0.08	1.70	-0.05	-0.67	-1.98
2	10	1	4	269	0.07	0.51	0.68	-0.55	-0.05	0.43	-0.62
2	10	2	8	327	0.1	-0.43	0.68	-0.12	-0.05	0.43	1.46
2	10	3	23	318	0.15	0.23	-0.08	-0.23	-0.05	0.43	0.69
2	10	4	3	346	0.12	-0.04	0.68	-0.74	-0.76	0.43	-0.31
2	10	5	9	271	0.17	0.14	0.73	0.34	-0.05	0.43	1.57
2	10	6	10	273	0.12	0.92	0.68	-0.54	-0.05	0.43	0.74
2	10	7	9	299	0.11	0.93	-0.08	-0.36	-0.05	0.43	0.66
2	10	8	6	307	0.08	-0.01	0.68	-0.52	-0.05	0.43	0.38
2	10	9	8	289	0.15	0.67	-0.08	-0.27	-0.05	0.43	-0.15
2	10	10	13	325	0.26	0.16	-0.08	0.00	-0.16	0.43	1.23
2	10	11	13	315	0.18	-0.46	0.68	-0.14	-0.05	0.43	0.64
2	10	12	10	259	0.12	0.83	0.68	-0.14	-0.05	0.43	0.32
2	10	13	7	234	0.06	-0.20	0.68	1.23	-0.05	0.43	0.35
2	10	14	12	260	0.16	0.74	0.72	0.08	-0.05	0.43	0.89
2	10	15	8	249	0.12	0.66	0.68	0.48	-0.14	0.43	0.81
2	10	16	13	268	0.09	0.04	0.68	0.49	-0.05	0.43	0.60
2	10	17	11	256	0.11	0.31	0.68	0.24	-0.05	0.43	-0.06
2	10	18	6	342	0.17	0.82	0.30	-0.76	-0.76	0.43	0.64
2	10	19	8	302	0.15	0.22	0.68	-0.33	-0.05	0.43	1.20
2	10	20	12	370	0.24	-0.28	0.75	-0.51	-0.76	0.43	0.84
2	10	21	19	281	0.11	0.29	0.68	-0.07	-0.05	0.43	0.64
2	10	22	9	290	0.13	-0.34	0.68	0.15	-0.05	0.43	0.20
2	10	23	13	351	0.11	0.21	0.68	-0.70	-0.76	0.43	0.66
2	11	1	10	411	0.22	-1.09	-0.93	0.23	-0.12	-0.67	-0.72
2	11	2	14	502	0.14	-1.13	-0.97	-1.01	-0.86	-0.67	-0.09
2	11	3	7	481	0.2	-0.81	-0.08	-1.20	-1.11	-0.36	-0.96
2	11	4	19	516	0.16	-0.95	-0.95	-1.13	-0.91	-0.67	-1.64
2	11	5	14	482	0.12	-1.58	-0.89	-0.16	-0.76	-0.67	-0.68
2	11	6	24	498	0.16	-1.36	-0.94	-0.85	-0.77	-0.67	-0.46
2	11	7	17	527	0.25	-1.29	-1.25	-1.06	-0.84	-0.61	-1.65
2	11	8	23	484	0.15	-1.01	-0.92	-0.84	-0.79	-0.67	-0.72
2	11	9	18	509	0.1	-1.67	-0.91	-0.78	-0.76	-0.67	-0.75
2	11	10	10	435	0.15	-0.94	-0.08	-0.59	-0.69	-0.67	-0.89
2	11	11	19	491	0.25	-1.54	-0.93	-0.50	-0.74	-0.67	-0.11
2	11	12	12	452	0.25	-1.47	-0.76	-0.12	-0.79	0.43	-0.55
2	11	13	8	445	0.17	-1.22	-0.08	-0.43	-0.67	-0.67	-0.19
2	11	14	8	427	0.08	-0.60	-0.89	-0.75	-0.05	-0.67	-0.70
2	11	15	17	466	0.16	-0.52	-0.98	-0.84	-0.76	-0.67	-0.67
2	11	16	17	511	0.16	-1.09	-1.00	-1.06	-0.88	-0.67	-1.12
2	11	17	8	437	0.08	-0.69	-0.08	-0.75	-0.76	-0.67	-0.48
2	11	18	9	465	0.09	-1.33	-0.08	-0.78	-0.76	-0.67	-0.56
2	11	19	10	525	0.18	-1.69	-1.19	-1.13	-0.90	-0.67	-0.23
2	11	20	30	473	0.29	-1.06	-0.87	-0.89	-0.79	0.43	-0.61
2	11	21	6	450	0.1	-1.62	-0.08	-0.06	-0.76	-0.67	-0.70
2	11	22	18	457	0.26	-1.05	-0.89	-0.10	-0.68	-0.67	-0.16
2	11	23	12	453	0.09	-1.43	-0.08	-0.29	-0.76	-0.67	-1.02
2	12	1	25	397	0.2	-0.09	0.68	-0.76	-0.70	-0.67	0.48
2	12	2	13	394	0.18	-0.08	0.68	-0.81	-0.79	-0.67	-0.16
2	12	3	13	413	0.17	0.17	0.68	-1.08	-0.87	-0.67	0.72
2	12	4	21	474	0.12	-0.56	0.68	-1.19	-1.10	-0.67	0.96
2	12	5	15	442	0.11	-0.32	0.68	-1.19	-1.09	-0.67	0.31
2	12	6	12	458	0.26	-0.88	0.68	-0.73	-0.70	-0.67	1.24
2	12	7	17	405	0.25	-0.57	0.68	-0.86	-0.90	0.43	0.44
2	12	8	23	421	0.14	0.08	-0.08	-0.77	-0.76	-0.67	0.62
2	12	9	24	468	0.16	-0.66	-0.08	-1.13	-0.98	-0.67	0.37
2	12	10	6	433	0.13	-0.26	-0.08	-0.91	-0.88	-0.67	-0.11
2	12	11	8	449	0.09	-0.21	0.84	-0.78	-0.76	-0.67	1.57
2	12	12	10	454	0.22	-0.75	0.72	-0.93	-0.94	0.43	1.25
2	12	13	19	428	0.17	-0.83	-0.08	-0.26	-0.68	-0.67	0.55
2	12	14	8	506	0.12	-0.51	1.05	-1.19	-1.11	-0.67	1.36
2	12	15	9	443	0.16	-0.77	-0.08	-0.08	-0.76	-0.67	1.22
2	12	16	21	441	0.19	-0.46	-0.08	-0.80	-0.62	-0.67	0.85
2	12	17	21	436	0.14	-0.85	0.68	-0.87	-0.76	-0.67	0.44
2	12	18	27	492	0.15	-0.84	-0.08	-1.17	-1.05	-0.67	0.90
2	12	19	9	495	0.15	-0.81	-0.08	-0.91	-0.88	-0.67	1.48
2	12	20	19	431	0.24	-0.99	0.68	-0.80	-0.78	-0.67	-0.34
2	12	21	16	387	0.2	-0.62	0.68	-0.11	-0.67	-0.67	0.49
2	12	22	10	477	0.19	-0.79	-0.08	-1.19	-1.11	0.43	0.74
2	12	23	27	415	0.13	-0.11	0.76	-0.76	-0.76	-0.67	1.01
2	13	1	10	138	0.21	0.94	0.68	0.67	1.37	0.43	-0.06
2	13	2	14	131	0.21	2.10	0.68	0.25	1.37	0.43	0.77
2	13	3	4	198	0.02	0.61	0.68	0.45	1.37	-0.67	0.69
2	13	4	2	47	0.19	1.96	1.49	2.29	1.37	0.43	1.36
2	13	5	8	143	0.08	1.28	1.11	0.45	1.37	-0.67	1.36
2	13	6	4	98	0.2	0.98	2.30	0.73	1.37	0.43	0.00
2	13	7	10	155	0.13	1.55	0.68	-0.28	1.37	0.43	0.33
2	13	8	24	151	0.12	1.45	0.68	0.22	1.37	0.43	0.72
2	13	9	5	164	0.11	1.30	-0.08	0.40	1.37	0.43	0.73
2	13	10	12	176	0.12	1.08	0.68	-0.15	1.37	0.43	0.72
2	13	11	10	94	0.13	1.39	2.30	0.44	1.37	0.43	1.23
2	13	12	12	154	0.12	1.75	0.68	0.36	1.37	-0.67	0.29
2	13	13	4	174	0.1	0.56	0.68	0.82	1.37	-0.67	0.19
2	13	14	18	153	0.27	0.89	0.75	0.30	1.37	0.43	1.33
2	13	15	10	172	0.22	0.58	0.77	0.34	1.37	0.43	0.47
2	13	16	8	177	0.08	0.88	0.68	0.45	1.37	-0.67	1.41
2	13	17	2	125	0.09	2.71	0.68	0.34	1.37	-0.67	0.67
2	13	18	8	95	0.07	1.62	2.30	0.45	1.37	-0.67	1.41
2	13	19	7	173	0.07	1.27	0.68	0.45	1.37	-0.67	0.74
2	13	20	7	128	0.1	1.78	0.68	0.38	1.37	0.43	-0.02
2	13	21	16	111	0.2	1.15	2.30	0.59	1.37	-0.67	0.18
2	13	22	10	152	0.11	1.78	0.68	0.46	1.37	-0.67	0.90
2	13	23	4	162	0.07	1.30	0.68	0.46	1.37	-0.67	0.05
2	14	1	46	475	0.08	-0.74	-0.89	-0.78	-0.76	-0.67	0.52
2	14	2	36	469	0.16	-0.24	-0.94	-0.83	-0.76	-0.67	0.94
2	14	3	19	439	0.18	-0.52	-0.95	-0.74	-0.05	-0.67	0.60
2	14	4	13	459	0.15	0.21	-1.04	-0.93	-0.76	-0.67	0.60
2	14	5	24	501	0.14	-0.59	-1.00	-1.16	-0.89	-0.67	0.87
2	14	6	44	480	0.13	-0.63	-0.93	-0.78	-0.76	-0.67	0.98
2	14	7	35	493	0.13	-0.68	-0.91	-1.17	-0.98	-0.67	0.42
2	14	8	19	451	0.1	-0.20	-0.95	-0.74	-0.76	-0.67	-0.06
2	14	9	19	483	0.14	-0.72	-1.03	-0.93	-0.85	-0.67	-0.07
2	14	10	34	488	0.13	-0.61	-1.27	-0.93	-0.76	-0.67	0.54
2	14	11	11	455	0.18	-0.85	-0.89	-0.17	-0.69	-0.67	0.45
2	14	12	37	462	0.1	-0.26	-0.91	-0.85	-0.76	-0.67	0.39
2	14	13	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	14	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	15	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	16	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	17	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	18	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	19	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	20	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	21	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	22	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	15	1	13	341	0.17	0.19	-0.92	-0.52	-0.05	0.43	-0.93
2	15	2	16	361	0.16	0.24	0.68	-0.91	-0.45	-0.67	-1.62
2	15	3	5	456	0.11	-0.29	-0.89	-0.70	-0.76	-0.67	-1.12
2	15	4	26	461	0.2	-0.35	-0.08	-1.04	-0.81	-0.67	-1.59
2	15	5	16	444	0.3	-0.40	0.68	-1.15	-0.94	-0.61	-1.48
2	15	6	15	486	0.08	-0.50	-0.89	-0.92	-0.76	-0.67	-1.66
2	15	7	5	203	0.2	0.69	-1.12	-0.43	1.37	-0.23	-1.58
2	15	8	9	426	0.33	-0.40	-0.75	-0.77	-0.68	0.43	-1.23
2	15	9	14	369	0.19	0.28	-0.92	-0.38	-0.05	-0.67	-0.90
2	15	10	11	464	0.07	-0.06	-0.89	-0.82	-0.76	-0.67	-1.66
2	15	11	5	460	0.11	0.54	-0.89	-0.88	-0.76	-0.67	-1.70
2	15	12	7	410	0.17	0.51	-0.08	-0.92	-0.76	-0.67	-1.57
2	15	13	17	352	0.15	0.35	-0.89	-0.72	-0.13	0.43	-1.66
2	15	14	11	420	0.08	0.02	-1.03	-0.67	-0.05	-0.67	-1.65
2	15	15	13	396	0.13	0.43	-0.89	-0.71	-0.05	-0.67	-1.63
2	15	16	4	379	0.23	0.75	-0.69	-0.65	-0.76	0.15	-0.67
2	15	17	7	322	0.21	0.47	0.24	-0.72	-0.66	0.43	-1.52
2	15	18	25	343	0.22	0.36	-0.08	-0.63	-0.05	-0.67	-1.53
2	15	19	15	496	0.09	-0.38	-1.04	-1.11	-0.76	-0.67	-1.63
2	15	20	14	380	0.3	0.03	-0.08	-0.55	-0.46	-0.67	-0.82
2	15	21	14	293	0.21	0.50	-0.08	-0.67	-0.05	0.43	-1.14
2	15	22	10	353	0.17	0.20	0.68	-0.81	-0.48	-0.67	-0.92
2	15	23	11	432	0.09	-0.35	-0.89	-0.80	-0.05	-0.67	-1.61
2	16	1	8	58	0.25	0.04	2.30	1.70	-0.05	2.64	-1.20
2	16	2	1	33	0	1.25	2.30	1.20	1.37	2.64	-0.94
2	16	3	13	90	0.39	0.07	2.30	0.48	-0.05	2.64	-0.99
2	16	4	5	230	0.11	0.87	-0.89	-0.67	-0.33	2.64	-1.64
2	16	5	11	163	0.15	-0.13	0.68	0.38	-0.05	2.64	-0.78
2	16	6	4	75	0.04	0.84	-0.89	1.20	1.37	2.64	-0.83
2	16	7	17	207	0.2	-0.62	0.68	0.36	-0.76	2.64	-1.01
2	16	8	20	135	0.14	-0.24	0.98	0.51	-0.05	2.64	-1.10
2	16	9	9	132	0.15	1.33	-0.08	-0.65	-0.29	2.64	-1.52
2	16	10	8	74	0.2	0.95	0.39	1.10	1.37	2.64	-0.65
2	16	11	8	284	0.05	-0.76	-0.08	0.51	-0.76	2.64	-1.16
2	16	12	5	83	0.04	0.78	-0.89	1.20	1.37	2.64	-0.43
2	16	13	5	105	0.07	-0.44	0.68	0.49	-0.05	2.64	-2.23
2	16	14	4	136	0.09	1.24	0.68	-0.64	-0.05	2.64	-0.87
2	16	15	2	106	0.03	1.56	0.68	-0.71	-0.76	2.64	-1.71
2	16	16	9	120	0.06	-0.01	0.68	1.23	-0.05	2.64	-0.72
2	16	17	7	67	0.21	0.47	0.41	1.49	1.37	2.64	-0.97
2	16	18	7	213	0.33	0.07	0.57	-0.21	-0.05	2.64	-0.36
2	16	19	5	160	0.15	0.43	0.68	0.62	-0.05	2.64	0.17
2	16	20	6	320	0.13	-0.58	0.68	-0.26	-0.76	2.64	-0.44
2	16	21	3	85	0.06	0.38	0.68	2.29	-0.05	2.64	-0.61
2	16	22	6	345	0.1	-0.82	-0.08	0.21	-0.76	2.64	-0.79
2	16	23	5	110	0.06	1.39	0.68	-0.67	-0.05	2.64	-1.61
2	17	1	6	440	0.09	-0.58	-1.27	0.12	-0.05	-0.67	1.29
2	17	2	10	416	0.13	-0.58	-0.93	-0.18	-0.05	-0.67	0.64
2	17	3	12	417	0.09	-0.24	-0.92	0.15	-0.05	-0.67	1.57
2	17	4	9	349	0.12	0.29	-0.93	0.23	-0.05	-0.67	-0.11
2	17	5	5	309	0.28	0.52	-0.89	0.35	-0.05	0.43	0.52
2	17	6	16	385	0.13	0.31	-0.89	-0.45	-0.05	-0.67	0.41
2	17	7	23	372	0.09	0.31	-0.91	0.03	-0.05	-0.67	0.86
2	17	8	21	403	0.09	-0.15	-1.27	0.09	-0.05	-0.67	0.85
2	17	9	14	366	0.13	-0.37	-0.92	0.45	-0.05	-0.67	-0.31
2	17	10	11	419	0.13	-0.26	-0.89	-0.36	-0.05	-0.67	1.02
2	17	11	13	389	0.11	-0.35	-0.95	0.02	-0.05	-0.67	0.00
2	17	12	17	392	0.06	-0.26	-0.89	0.05	-0.05	-0.67	0.62
2	17	13	4	362	0.17	0.15	-1.08	0.01	-0.05	-0.67	-0.67
2	17	14	9	424	0.17	0.37	-0.89	-0.29	-0.76	-0.67	0.90
2	17	15	12	395	0.09	-0.06	-1.27	-0.09	-0.05	-0.67	0.37
2	17	16	18	399	0.11	-0.17	-0.89	0.09	-0.05	-0.67	1.05
2	17	17	15	373	0.09	-0.59	-0.89	0.45	-0.05	-0.67	-0.58
2	17	18	11	357	0.09	0.20	-0.89	0.47	-0.05	-0.67	0.68
2	17	19	15	398	0.26	0.98	-0.89	-0.41	-0.57	-0.67	0.49
2	17	20	20	367	0.08	0.25	-0.91	0.05	-0.05	-0.67	0.46
2	17	21	3	438	0.08	-0.60	-0.89	0.04	-0.76	-0.67	0.48
2	17	22	6	429	0.08	-0.22	-0.89	-0.07	-0.76	-0.67	0.68
2	17	23	15	382	0.08	-0.35	-0.92	0.45	-0.05	-0.67	0.47
2	18	1	4	430	0.06	-0.12	2.30	-0.78	-0.76	-0.67	0.52
2	18	2	5	199	0.07	0.53	2.30	0.46	-0.05	0.43	0.62
2	18	3	4	276	0.04	-0.21	2.30	0.03	-0.05	-0.67	-0.48
2	18	4	1	225	0	0.74	2.30	-0.19	-0.05	-0.67	-0.79
2	18	5	4	340	0.08	0.42	2.30	-0.56	-0.76	-0.67	-0.64
2	18	6	16	286	0.15	0.56	2.30	0.00	-0.05	-0.67	1.49
2	18	7	5	242	0.04	0.41	2.30	0.45	-0.05	-0.67	0.35
2	18	8	7	187	0.09	0.61	2.30	0.31	-0.05	0.43	1.57
2	18	9	8	200	0.15	0.62	2.30	0.08	-0.05	0.43	0.04
2	18	10	5	226	0.04	0.41	2.30	0.05	-0.05	0.43	0.69
2	18	11	8	463	0.07	0.24	2.30	-0.78	-0.76	-0.67	1.30
2	18	12	9	522	0.08	-0.14	2.30	-1.19	-1.11	-0.67	1.33
2	18	13	2	115	0.08	1.74	2.30	1.23	-0.05	-0.67	-0.03
2	18	14	6	274	0.18	0.31	2.30	-0.06	-0.05	-0.67	0.15
2	18	15	3	229	0.06	0.55	2.30	-0.06	-0.05	0.43	1.14
2	18	16	10	254	0.13	0.67	2.30	0.22	-0.05	-0.67	0.87
2	18	17	4	407	0.13	0.00	2.30	-0.78	-0.76	0.43	0.94
2	18	18	7	246	0.08	0.16	2.30	-0.04	-0.05	0.43	1.57
2	18	19	4	339	0.04	-0.04	2.30	0.05	-0.76	-0.67	0.04
2	18	20	3	217	0.01	0.65	2.30	0.03	-0.05	0.43	0.69
2	18	21	3	476	0.14	-0.52	2.30	-0.45	-0.76	-0.67	1.29
2	18	22	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	19	1	6	391	0.18	0.52	-0.89	-0.34	-0.64	0.43	0.98
2	19	2	6	383	0.07	0.68	-0.89	-0.56	-0.76	0.43	0.10
2	19	3	21	470	0.12	-1.02	-0.91	-0.76	-0.76	0.43	0.53
2	19	4	11	348	0.2	0.08	-1.03	-0.10	-0.05	0.43	0.14
2	19	5	10	446	0.15	-1.12	-0.81	-0.30	-0.76	0.43	0.01
2	19	6	14	350	0.08	0.07	-0.89	0.05	-0.05	0.43	0.74
2	19	7	8	388	0.27	-0.53	-0.38	-0.56	-0.40	0.43	-0.28
2	19	8	7	467	0.11	-1.18	-0.89	-0.29	-0.76	0.43	0.88
2	19	9	6	425	0.09	-0.49	-0.89	-0.23	-0.76	0.43	0.58
2	19	10	9	406	0.1	-0.18	-0.08	-0.75	-0.76	0.43	0.63
2	19	11	10	400	0.16	-0.36	-0.89	-0.71	-0.05	0.43	0.33
2	19	12	9	347	0.21	-0.22	-0.89	0.49	-0.13	0.43	0.63
2	19	13	8	497	0.1	-0.83	-0.89	-1.19	-1.11	0.43	0.44
2	19	14	16	378	0.12	-0.25	-1.03	-0.12	-0.05	0.43	0.74
2	19	15	16	418	0.15	-0.73	-0.08	-0.41	-0.76	0.43	0.82
2	19	16	22	448	0.15	-0.21	-0.94	-0.82	-0.76	0.43	0.77
2	19	17	13	485	0.15	-0.74	-0.95	-0.83	-0.76	0.43	1.20
2	19	18	13	447	0.14	-0.52	-0.98	-0.75	-0.76	0.43	0.17
2	19	19	10	412	0.21	-0.48	-0.81	-0.43	-0.05	0.43	1.25
2	19	20	11	355	0.16	-0.61	-0.89	0.20	-0.05	0.43	-0.14
2	19	21	9	390	0.1	-0.66	-0.89	-0.14	-0.05	0.43	0.65
2	19	22	10	401	0.15	-0.52	-0.93	0.13	-0.05	0.43	1.40
2	19	23	11	487	0.12	-0.83	-1.27	-0.86	-0.76	0.43	0.75
2	20	1	5	73	0.02	-0.13	0.68	1.70	1.37	0.43	-2.46
2	20	2	3	69	0.02	-0.02	0.68	2.29	1.37	0.43	-1.98
2	20	3	3	65	0.19	-0.83	1.22	3.07	-0.05	-0.67	-2.46
2	20	4	7	79	0.03	0.82	2.30	1.21	1.37	0.43	-0.92
2	20	5	5	62	0.03	0.13	0.68	2.29	1.37	0.43	-2.32
2	20	6	3	80	0.04	-0.03	0.68	1.70	1.37	0.43	-2.19
2	20	7	3	71	0.03	0.47	2.30	1.70	1.37	0.43	-0.79
2	20	8	2	72	0	0.22	0.68	1.70	1.37	0.43	-2.46
2	20	9	12	35	0.14	0.15	2.30	2.29	1.37	0.43	-2.29
2	20	10	4	44	0.06	-0.03	0.68	3.07	1.37	0.43	-2.25
2	20	11	11	42	0.07	0.15	2.30	1.70	1.37	0.43	-2.39
2	20	12	1	31	0	0.57	2.30	3.30	1.37	0.43	-1.25
2	20	13	5	63	0.08	-0.29	0.68	2.29	1.37	0.43	-2.35
2	20	14	6	66	0.05	0.39	2.30	1.70	1.37	0.43	-1.23
2	20	15	1	68	0	0.31	0.68	2.29	1.37	0.43	-1.98
2	20	16	3	78	0.05	1.06	2.30	1.23	1.37	0.43	-0.62
2	20	17	4	27	0.05	0.19	2.30	3.07	1.37	0.43	-2.25
2	20	18	0	NA	NA	NA	NA	NA	NA	NA	NA
2	20	19	0	NA	NA	NA	NA	NA	NA	NA	NA
2	20	20	0	NA	NA	NA	NA	NA	NA	NA	NA
2	20	21	0	NA	NA	NA	NA	NA	NA	NA	NA
2	20	22	0	NA	NA	NA	NA	NA	NA	NA	NA
2	20	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	21	1	4	504	0.17	0.72	-0.89	-0.66	-0.76	2.64	1.36
2	21	2	5	245	0.17	0.97	0.68	-0.65	-0.62	2.64	1.03
2	21	3	6	313	0.16	-0.20	0.68	-0.40	-0.52	2.64	0.41
2	21	4	10	338	0.18	0.27	-0.89	-0.34	-0.05	2.64	0.32
2	21	5	4	375	0.13	0.16	-0.49	0.04	-0.76	2.64	0.66
2	21	6	6	298	0.15	1.35	-0.08	-0.45	-0.76	2.64	0.94
2	21	7	10	526	0.21	-0.31	-1.08	-0.88	-0.76	2.64	1.28
2	21	8	11	490	0.15	-0.83	-0.89	-0.25	-0.76	2.64	-0.33
2	21	9	6	393	0.19	1.39	-0.89	-0.56	-0.76	2.64	-0.04
2	21	10	8	359	0.11	0.49	-0.89	-0.39	-0.05	2.64	0.95
2	21	11	6	295	0.1	0.19	-0.08	-0.29	-0.05	2.64	0.75
2	21	12	7	376	0.09	-0.05	-0.89	-0.19	-0.05	2.64	0.77
2	21	13	18	205	0.2	0.58	0.68	-0.13	-0.13	2.64	0.65
2	21	14	6	186	0.18	0.55	0.55	0.01	-0.05	2.64	1.50
2	21	15	6	328	0.14	0.27	-0.89	-0.73	-0.05	2.64	-0.76
2	21	16	2	215	0.1	0.75	-0.08	-0.46	-0.05	2.64	-0.35
2	21	17	5	408	0.12	-0.53	-0.08	-0.29	-0.76	2.64	0.19

Now let’s check the compression summary for HVT (map C). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)

segmentLevel	noOfCells	noOfCellsBelowQuantizationError	percentOfCellsBelowQuantizationErrorThreshold	parameters
1	23	0	0	n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2	508	434	0.85	n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 85% of the cells have hit the quantization threshold error in level 2

Let’s plot the Voronoi tessellation for layer 2 (map C)

muHVT::plotHVT(map_C,
        line.width = c(0.4,0.2), 
        color.vec = c("#141B41","#0582CA"),
        centroid.size = 0.1,
        maxDepth = 2)

Figure 13: The Voronoi Tessellation for layer 2 (map C) shown for the 100 cells in the dataset ’computers’ at level 2

Heat Maps

Now let’s plot all the features for each cell at level two as a heatmap for better visualization.


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "n",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 14: The Voronoi Tessellation with the heat map overlaid for features No. of entities in each cell


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "price",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 15: The Voronoi Tessellation with the heat map overlaid for features price in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "speed",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 16: The Voronoi Tessellation with the heat map overlaid for features speed in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "hd",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 17: The Voronoi Tessellation with the heat map overlaid for features hd in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ram",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 18: The Voronoi Tessellation with the heat map overlaid for features ram in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "screen",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 19: The Voronoi Tessellation with the heat map overlaid for features screen in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ads",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 20: The Voronoi Tessellation with the heat map overlaid for features ads in the ’computers’ dataset

We now have the set of maps (map A, map B & map C) which will be used to predict which map and cell each test record is assigned to, but before that lets view our test dataset

6 Prediction on Test Data

Now once we have built the model, let us try to predict using our test dataset which cell and which layer each point belongs to.

Raw Testing Dataset

The testing dataset includes the following columns:

Price: The price of 486 PCs in US dollars.
Speed: The clock speed of the PCs in MHz.
HD: The size of the hard drive in MB.
RAM: The size of RAM in MB.
Screen: The size of the screen in inches.
Ads: The number of listings for 486 PCs’ prices for each month.

Let’s have a look at our randomly selected test dataset containing 1253 datapoints.

Table(head(testComputers_data))

Row.No	price	speed	hd	ram	screen	ads
3	1595	25	170	4	15	94
4	1849	25	170	8	14	94
7	1720	25	170	4	14	94
10	2575	50	210	4	15	94
11	2195	33	170	8	15	94
14	2295	25	245	8	14	94

The predictLayerHVT function is used to score the test data using the predictive set of maps. This function takes an input - a test data and a set of maps (map A, map B, map C).

Now, Let us understand the predictLayerHVT function.

predictLayerHVT(data,
                map_A,
                map_B,
                map_C,
                mad.threshold = 0.2,
                normalize = T, 
                distance_metric="L1_Norm",
                error_metric="max",
                child.level = 1, 
                line.width = c(0.6, 0.4, 0.2),
                color.vec = c("#141B41", "#6369D1", "#D8D2E1"),
                yVar= NULL,
                ...)

Each of the parameters of predictLayerHVT function has been explained below:

data - A dataframe containing the test dataset. The variables from this dataset can also be used to overlay as heatmap.
map A - A list of hvt.results.model obtained from HVT function while performing hierarchical vector quantization on train data.
map B - A list of hvt.results.model obtained from HVT function while performing hierarchical vector quantization on data with novelty which was obtained as a result of removeNovelty function.
map C - A list of hvt.results.model obtained from HVT function while performing hierarchical vector quantization on data without novelty obtained as a result of removeNovelty function.
child.level - A number indicating the layer for which the heat map is to be plotted.(Only used if hmap.cols is not NULL).It specifies the specific layer within the hierarchical structure of the model.
mad.threshold - A numeric values indicating the permissible Mean Absolute Deviation.
normalize - A logical value indicating if the columns in your dataset should be normalized. Default value is TRUE, which means the columns will be normalized.
distance_metric - The distance metric can be ’Euclidean” or “Manhattan”. Euclidean is selected by default.
error_metric - The error metric can be “mean” or “max”. mean is selected by default.
yVar - Name of the dependent variable(s).
... - color.vec and line.width can be passed from here.

The function predicts based on the HVT maps - map A, map B and map C, constructed using HVT function. For each test record, the function will assign that record to Layer1 or Layer2. Layer1 contains the cell ids from map A and Layer 2 contains cell ids from map B (novelty map) and map C (map without novelty).

Prediction Algorithm

The prediction algorithm recursively calculates the distance between each point in the test dataset and the cell centroids for each level. The following steps explain the prediction method for a single point in the test dataset:

Calculate the distance between the point and the centroid of all the cells in the first level.
Find the cell whose centroid has minimum distance to the point.
Check if the cell drills down further to form more cells.
If it doesn’t, return the path. Or else repeat steps 1 to 4 till we reach a level at which the cell doesn’t drill down further.

Note : The prediction algorithm will not work if some of the variables used to perform quantization are missing. In the test dataset, we should not remove any features

Let’s see which cell and layer each point belongs to.For the sake of brevity, we will only show the first 10 rows.


validation_data <- testComputers
new_predict <- predictLayerHVT(
    data=validation_data,
    map_A,
    map_B,
    map_C,
    normalize = T
  )

prediction_output <- head(new_predict,7)
row_indices <- c(287, 397, 873)
new_df <- new_predict[row_indices, ]
Predictions <- rbind(prediction_output,new_df)
row.names(Predictions) <- NULL

Predictions%>% head(100) %>%
  as.data.frame() %>%
  Table(scroll = T, limit = 20)

Row.Number	Layer1.Cell.ID	Layer2.Cell.ID
1	A24	C478
2	A143	C404
3	A29	C478
4	A237	C404
5	A223	C404
6	A162	C404
7	A215	C404
287	A440	B1
397	A440	B1
873	A438	B1

Note: From the above table, we can see that 343rd observation from the test data having Cell.ID as A399 have been identified as novelty and is mapped to 1st(B1) novelty cell in Layer2.Cell.ID column .Similarly, 367th,423rd……. test record having Cell.ID as A399 is correctly identified as novelty and gets mapped to B1 novelty cell in the Layer2.Cell.ID column respectively

summary_list <- map_A[[3]]

  train_colnames <- names(summary_list[["nodes.clust"]][[1]][[1]])
  scaled_test_data <- scale(
     testComputers[, train_colnames],
      center = summary_list$scale_summary$mean_data[train_colnames],
      scale = summary_list$scale_summary$std_data[train_colnames])

testComputers <- scaled_test_data
data1 <- data.frame(testComputers)
data1$Row.No <- row.names(testComputers_data)
data1 <- data1 %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
colnames(data1) <- c("Row.No","price_act","speed_act","hd_act","ram_act","screen_act","ads_act")
Layer2.Cell.ID <- new_predict$Layer2.Cell.ID
combined <- cbind(data1,Layer2.Cell.ID)
combined <- combined %>%
  mutate(Cell.ID = gsub("[BC]", "", Layer2.Cell.ID))
mapC_summary <- map_C[[3]]$summary

df1 <- combined
df1$Cell.ID <- as.numeric(df1$Cell.ID)
df2 <- mapC_summary
merged_df <- merge(df1, df2, by = "Cell.ID", all.x = TRUE)
sorted_df <- merged_df[order(merged_df$Row.No), ]
sorted_df <- sorted_df %>% select(price_act,speed_act,hd_act,ram_act,screen_act,ads_act,Layer2.Cell.ID,price,speed,hd,ram,screen,ads)
sorted_df$Row.No <- testComputers_data$Row.No

sorted_df <- sorted_df%>% dplyr::select(Row.No,price_act,speed_act,hd_act,ram_act,screen_act,ads_act,Layer2.Cell.ID,price,speed,hd,ram,screen,ads)
colnames(sorted_df) <- c("Row.No","price_act","speed_act","hd_act","ram_act","screen_act","ads_act","Layer2.Cell.ID","price_pred","speed_pred","hd_pred","ram_pred","screen_pred","ads_pred")
sorted_df$diff <- rowMeans(abs(sorted_df[, c("price_act","speed_act","hd_act","ram_act","screen_act","ads_act")] - sorted_df[, c("price_pred","speed_pred","hd_pred","ram_pred","screen_pred","ads_pred")]))
rownames(sorted_df) <- NULL
options(scipen = 999)
sorted_df %>% head(100) %>%
  as.data.frame() %>%
  Table(scroll = T, limit = 10)

Row.No	price_act	speed_act	hd_act	ram_act	screen_act	ads_act	Layer2.Cell.ID	price_pred	speed_pred	hd_pred	ram_pred	screen_pred	ads_pred	diff
3	-1.0785679	-1.2710382	-0.9472574	-0.7589986	0.4307274	-1.7213059	C478	-1.1829114	-0.7922801	-0.6999883	-0.7557764	-0.5234546	-0.6945646	0.4690860
4	1.1382448	-0.0817113	-0.7914378	-0.7589986	2.6404120	-1.7213059	C360	0.2356188	-0.4603485	-0.3744382	-0.4107768	2.6404120	0.5948051	0.7270993
7	1.3374115	-0.0817113	0.1512706	1.3676416	0.4307274	-1.1163339	C201	1.5745305	0.4514353	-0.2367672	0.2737663	-0.3026593	-1.3041998	0.5289052
10	-1.0629809	0.6794579	-0.7758559	-0.7589986	-0.6741149	-0.5382496	C478	-1.1829114	-0.7922801	-0.6999883	-0.7557764	-0.5234546	-0.6945646	0.3296222
11	-0.6455967	0.6794579	0.4473277	-0.0501185	-0.6741149	-0.5382496	C306	-0.7716661	0.5467115	0.2755179	-0.2228202	-0.1702066	-0.7871317	0.2260196
14	0.9286867	-0.8904536	2.2859986	2.7854017	0.4307274	-0.5382496	C55	1.4810668	0.0181922	2.3175726	2.6287019	0.1225346	-0.2170130	0.3797882
15	1.0845564	-0.8904536	2.2859986	2.7854017	0.4307274	-0.5382496	C55	1.4810668	0.0181922	2.3175726	2.6287019	0.1225346	-0.2170130	0.3538099
19	-0.8984519	0.6794579	0.0266149	-0.0501185	-0.6741149	-0.5382496	C306	-0.7716661	0.5467115	0.2755179	-0.2228202	-0.1702066	-0.7871317	0.2389879
22	-1.0716404	0.6794579	0.4940736	-0.0501185	-0.6741149	-0.5382496	C306	-0.7716661	0.5467115	0.2755179	-0.2228202	-0.1702066	-0.7871317	0.2627947
24	-1.3660608	-0.8904536	0.0421969	-0.7589986	0.4307274	-0.5382496	C478	-1.1829114	-0.7922801	-0.6999883	-0.7557764	-0.5234546	-0.6945646	0.3562045

hist(sorted_df$diff, breaks = 20, col = "blue", main = "Mean Absolute Difference", xlab = "Difference")

Figure 21: Mean Absolute Difference

muHVT: Predicting Cells with Layers using predictLayerHVT

Zubin Dowlaty, Srinivasan Sudarsanam, Somya Shambhawi

2023-06-23

1 Abstract

2 Example: muHVT with the Personal Computer dataset

3 Map A : Base Compressed Map

3.0.1 Iteration 1:

3.0.2 Iteration 2:

4 Map B : Compressed Novelty Map

4.1 Voronoi Tessellation with highlighted novelty cell

5 Map C : Compressed Map without Novelty

5.0.1 Iteration 1:

5.0.2 Iteration 2:

6 Prediction on Test Data

7 Executive Summary

8 References